 
\input amstex
\magnification=\magstep1
\hsize=16truecm
\TagsOnRight
\parskip=2pt plus 1pt
\define\f{\varphi}
\define\({\left(}
\define\){\right)}
\define\[{\left[}
\define\]{\right]}
\define\fb{{\bar\f}}
\define\e{\varepsilon}
\define\const{\text{\rm const.}\,}
 
\centerline{\bf THE LIMIT BEHAVIOUR OF ELEMENTARY SYMMETRIC}
\centerline{\bf POLYNOMIALS OF I.I.D. RANDOM VARIABLES}
\centerline{\bf WHEN THEIR ORDER TENDS TO INFINITY}
\medskip
\centerline{P\'eter Major}
\centerline{\it Mathematical
Institute of the Hungarian Academy of Sciences}
\centerline{\it and }
\centerline{\it Bolyai College of the E\"otv\"os Lor\'and University,
Budapest}
\bigskip\noindent{\narrower {\narrower{\it Abstract:}\/ Let
$\xi_1,\xi_2,\dots$ be a sequence of i.i.d.\ random variables, and
consider the elementary symmetric polynomial $S^{(k)}(n)$ of order
$k=k(n)$ of the first $n$ elements $\xi_1,\dots,\xi_n$ of this sequence.
We are interested in the limit behaviour of $S^{(k)}(n)$ with an
appropriate transformation if $\frac {k(n)}n\to\alpha$, $0<\alpha<1$.
Since $k(n)\to\infty$ as $n\to\infty$, the classical methods cannot be
applied in this case and new kind of results appear. We solve the
problem under some conditions which are satisfied in the generic case.
The proof is based on the saddle point method and a limit theorem for
sums of independent random vectors which may have some special interest
in itself.\par}\par}   \medskip
 
\beginsection 1. Introduction
 
In this paper
\plainfootnote{} {The author got support from the Hungarian OTKA
Foundation No. T26176.}
the following problem is investigated: Let $\xi_1,\dots,
\xi_n$ be i.i.d.\ random variables with some non-degenerate
distribution function $F(x)$, i.e.\ we assume that the distribution of
the random variables $\xi_j$, $j=1,\dots,n$ is not concentrated
in a single point. Define the elementary symmetric polynomials
$$
S^{(k)}(n)=S^{(k)}(n,\xi_1,\dots,\xi_n)=\sum_{1\le i_1<
i_2<\cdots<i_k\le n} \xi_{i_1}\cdots \xi_{i_k}. \tag1.1
$$
We are interested in the limit behaviour of the random variables
$S^{(k)}(n)$ if $n\to\infty$, $k=k(n)$, and $\alpha(n)\to \alpha^*$,
$P(\xi=0)<\alpha^*<1$, where $\alpha(n)=1-\dfrac{k(n)}n$. The expression
defined in (1.1) is a special $U$-statistic of order $k$.
 
The limit behaviour of $U$-statistics for fixed $k$ is fairly well
understood, (see e.g.\ [1]). These results imply in particular that if
$E\xi=0$, then for fixed $k$ the random variables $n^{-k/2}S^{(k)}(n)$
have a limit distribution which can be expressed by means of a $k$-fold
multiple Wiener integral. But in our case the number~$k=k(n)$ tends
to infinity simultaneously with $n$. Hence the classical results cannot
be applied, and a different kind of limit theorems appears. The problem
we discuss here was investigated in earlier papers in some special
cases (see~[2], [3] and [4]). In paper [3] a law of large numbers was
proved if the random variables $\xi_j$ are non-negative, and in
paper~[4] the limit behaviour of $S^{(k)}(n)$ was described in the
special case when $P(\xi_j=1)=P(\xi_j=-1)=1/2$. Paper~[2] contains a
generalization of paper~[4] when the distribution of $\xi_j$ is
concentrated in three point, 0 and $\pm1$, and $P(\xi_j=1)=P(\xi_j=-1)
=1/2P(\xi_j\neq0)$. But the method of this paper is not strong enough to
handle more general distributions.
 
The proof of the above papers was based on the saddle point method. In
this paper also this method is applied. Several technical difficulties
had to be overcome to make this method work in the general case. It
shows a strong similarity with the technique applied in the theory of
large deviations.
 
We also want to understand whether the limit distribution of the
appropriately transformed statistics $S^{(k)}(n)$ shows some
universality, i.e.\ whether it depends only on
$\alpha^*=\lim\limits_{n\to\infty} \alpha(n)$ or it strongly depends on
the sequence $k(n)$ and the distribution function $F(x)$ of the random
variables $\xi_j$. We prove that in the generic case, although the
normalization depends on $\alpha(n)$, the limit distribution depends
only on $\alpha^*$.
 
The investigation is based on the following observation. Define the
polynomial
$$
Z_n(x)=Z_n(x,\xi_1,\dots,\xi_n)=\prod_{j=1}^n (x+\xi_j).
$$
Then
$$
Z_n(x)=\sum_{k=1}^n S^{(k)}(n)x^{n-k},
$$
hence
$$
\aligned
S^{(k)}(n)&=\frac1{(n-k)!}\frac{d^{(n-k)}}{dx^{(n-k)}}Z_n(x)=
\frac1{2\pi i}\oint_{|\zeta|=r}
\frac{Z_n(\zeta)}{\zeta^{n-k+1}}\,d\zeta\\
&=\frac 1{2\pi} \int_{-\pi}^{\pi}\frac
{\prod\limits_{j=1}^n|re^{i\varphi}+\xi_j|} {r^{n-k}}\exp\left\{-
i(n-k)\varphi+ i\sum_{j=1}^n\text{\,arg\,}(r e^{i\varphi}+\xi_j)
\right\}\,d\varphi
\endaligned \tag1.2
$$
for arbitrary $r>0$. We investigate the expression $S^{(k)}(n)$ in the
form defined in (1.2). To handle this integral it is natural to choose
the constant $r$, the radius of the circle where the integration is
taken, in the way as the saddle point method suggests. Hence it is
natural to look for a point
$(r,\fb)=(r(\xi_1,\dots,\xi_n),\fb(\xi_1,\dots,\xi_n))$
where the partial derivatives of the (random) expression
$$
\sum_{j=1}^n\log|re^{i\varphi}+\xi_j|- (n-k)\log r
$$
disappear. In papers~[2], [3] and [4] such an approach was applied. We
shall slightly modify this method by looking for an approximative
solution, for an asymptotic but non-random approximation of the saddle
point. The laws of large numbers suggests that
$$
\sum_{j=1}^n\log|re^{i\varphi}+\xi_j|\sim nE \log|re^{i\varphi}+\xi|
=nH(r,\varphi)
$$
with
$$
H(r,\f)= H(z)=E\log|re^{i\varphi}+\xi|=
\dfrac12E\log\(r^2+\xi^2+2r\xi\cos\f\), \tag 1.3
$$
where $\xi$ is an $F$ distributed random variable, and $z=re^{i\f}$.
Because of the parity properties of the integral at the right-hand side
of (1.3) it is enough to look for the (asymptotic) saddle point for
$0\le \varphi\le\pi$, i.e.\ for a solution in the upper half-plane. We
will show that under general conditions there is a point $(r,\fb)$,
$\fb=\fb(r)$, such that the relations
$$
\frac{\partial}{\partial \f}\[H(r,\bar\varphi)-\alpha(n)\log
r\]=0,\quad \frac{\partial}{\partial r}\[H(r,\bar\varphi)-
\alpha(n)\log r\]=0
$$
hold. We rewrite these equations in the equivalent form
$$
\left.\frac{\partial}{\partial \f}H(r,\f)=0\right|_{\f=\fb},\quad
r\left.\frac{\partial}{\partial r}H(r,\f)\right|_{\f=\fb}=\alpha(n),
\tag1.4
$$
and also require that the solution $(r,\fb)$ satisfy the relation
$$
\fb\text{ is the place of maximum of } H(r,\f)\quad(\text{as a function
of $\varphi$, $0\le \varphi\le \pi$.}). \tag1.5
$$
Let us remark that the solution of the equation (1.4) (together with the
property (1.5) depends on $n$ through the function $\alpha(n)$. Although
this dependence on $n$ will turn out to be weak in the case when
$\lim\limits_{n\to\infty}\alpha(n)=\alpha^*$, we need to investigate
carefully the dependence of the solution on $n$. This problem will
appear first of all in Section~4, and in that Section we shall indicate
explicitly the dependence on the parameter $n$.
 
We shall prove under general conditions that the equation (1.4) has a
unique solution  $(r,\fb)$ $0\le \fb\le \pi$ which also satisfies
relation (1.5). This result enables us to give a good asymptotic
expression of formula (1.2) and to approximate $S^{(k)}(n)$ by a
function of sum of independent random vectors. In such a way the limit
behaviour of $S^{(k)}(n)$  with an appropriately normalization can be
described by means of a limit theorem for sums of independent random
vectors. Since some technical conditions appear in the formulation of
the results about the limit behaviour of $S^{(k)}(n)$ we formulate them
only in Section~2.
 
The limit theorem for sums of independent random vectors needed in this
paper may be interesting in itself. In this limit theorem such a limit
distribution appears whose coordinates are independent. This
independence is not because of some uncorrelatedness property of the
coordinates of the summands. It has a structural reason. It
appears, because the partial sums of such random vectors are considered
whose first coordinates take values in a non-compact and the second
coordinates in a compact space. (We consider such random vectors whose
first coordinates, the absolute value of  random complex numbers, take
their values in the real line, and the second coordinates, the angle of
these complex numbers, take their values in the unit circle.) Similar
results in more general spaces were proved in [6].
 
This paper consists of six sections. In Section~2 we explain the method
of the paper, formulate some technical results and the main theorems.
In Section~3 we prove that under general conditions the asymptotic
saddle point equation (1.4) together with relation (1.5) can be solved.
In Section~4 we give a good asymptotic approximation of $S^{(k)}(n)$ by
means of an expansion of the integrand in (1.2) around the solution of
the saddle point equation~(1.4). In Section~5 a limit theorem for sums
of independent vectors needed in this paper is proved. Finally in
Section~6 the main results of the paper are proved.
 
\beginsection 2. The strategy of the proof
 
Consider the function $H(r,\f)=H(z)$ defined in $(1.3)$. First we want
to prove that under general conditions for the distribution of $F(x)$
of the random variables $\xi_j$ the equation (1.4) has a unique
solution which also satisfies (1.5). In the proof we investigate the
differentials of the function $H(r,\f)$. In these calculations the
order of differentiation and expectation will be changed several times.
To legitimate such steps some conditions will be imposed on the
distribution of the distribution function $F(x)$.
 
It is simple to justify these calculations in the neighbourhood of such
points $z=re^{i\f}$ for which the number $z$ has a non-zero imaginary
part, i.e.\ for which $\f\neq0$ and $\f\neq\pi$. On the other hand, for
$\f=0$ or $\f=\pi$ such a calculation is allowed only under fairly
restrictive conditions. But we shall differentiate only in the
neighbourhood of a point which can appear as the solution of the
equation (1.4) with some $\alpha(n)$, therefore we have not to impose
too restrictive conditions. We shall formulate such a condition on
$F(x)$ which probably  can be weakened, but which is satisfied by all
``nice" distribution functions. To formulate this condition let us
introduce the functions
$$
K^\pm(r)=E\dfrac{\pm\xi}{(\xi\pm r)^2},\quad r>0    \tag2.1
$$
and  sets
$$
\Cal A^\pm=\{r\: \;r>0 \text{ and } K^\pm(r)\ge 0\},     \tag2.2
$$
where $\xi$ is an $F(x)$ distributed random variable. Let us remark that
the integral (2.1) is always meaningful, although the relation
$E\dfrac{\pm\xi}{(\xi\pm r)^2}=-\infty$ is possible, since the
integrands in these expressions have an upper bound depending only on
$r$. As later calculation will show, it is enough to justify the change
of order of expectation and differentiation only in a small
neighbourhood of the real numbers $r$, $r\in \Cal A^+\cup
\Cal A^-$.
 
We formulate the following property: \medskip\noindent
{\bf Property A.} {\it If $r\in \Cal A^+$, then there is a number
$h=h(r)>0$ such that the interval $(-r-h,-r+h)$ has zero $F$ measure.
If $r\in \Cal A^-$, then there is a number $h=h(r)>0$ such that the
interval $(r-h,r+h)$ has zero $F$ measure.
 
This property can be formulated in the following equivalent form. Let
$\bold\Sigma$ denote the support of the distribution of $\xi$, i.e.\ the
smallest closed set on the real line $\bold R^1$ such that
$P(\xi\in\bold\Sigma)=1$. (Such a set exists. See e.g.\  [5], Chapter~2,
Theorem~2.1.) Then for all $r\in \Cal A^+$
$d(r,-\bold\Sigma)>0$ and for all $r\in\Cal A^-$ $d(r,\bold\Sigma)>0$.}
\medskip
Property~A is less restrictive than it may seem in the first moment,
because the sets $\Cal A^\pm$ are small. Thus for instance,
$r\notin \Cal A^\pm$ if the distribution function $F$ has a non-zero
density function in a neighbourhood of the point $\mp r$, or more
generally if $F(\mp r+h)-F(\mp r)>C\eta^2$ or $F(\mp r)-F(\mp
r-h)>C\eta^2$ with some  $C>0$, $h>0$ and  $0<\eta<h$. Indeed,
$K^\pm(r)=-\infty$ in this case. Thus Property~A holds if for all $x$
$F(x+h)-F(x-h)\ge \const h^2$ or $F(x+h)-F(x-h)=0$ if
$h<h_0$. Here both $h_0$ and $\const$ may depend on~$x$. Let us also
remark that also Property~A holds if an $F$ distributed $\xi$ random
variable is symmetrically distributed, since the sets $\Cal A^\pm$ are
empty in this case. Indeed, in this case
$$
K^\pm(r)=E\frac{\pm\xi}{(r\pm\xi)^2}=\frac12E\(\frac{\pm\xi}{(r\pm\xi)^2}
+\frac{\mp\xi}{(r\mp\xi)^2}\)=-E\frac{2r\xi^2}{((r^2-\xi^2)^2}<0
$$
for all $r>0$.
 
We also assume that
$$
E|\xi|<\infty,\quad\text{and}\quad E\frac1{|\xi|}I(\xi\neq0)<\infty
\tag 2.3
$$
We shall assume in the sequel that the distribution function $F$
satisfies Property~A and formula (2.3). The following three lemmas which
will be proved in Section~3 imply that if $P(\xi=0)<\alpha(n)<1$, then
the equation (1.4) has a unique solution which satisfies (1.5).
 
\medskip\noindent{\bf Lemma 1.} {\it Fix some $r>0$ and consider the
function $H(r,\f)$, defined in formula $(1.3)$ as a function of $\f$,
$0\le \f\le \pi$. (The function $H(r,\f)$ can also take the value
$-\infty$ in the end points 0 and $\pi$.) The function $H$ has a unique
maximum at a value $\fb=\fb(r)$ defined by the formula
$$
\fb(r)=\left\{
\alignedat2
&0 \quad&&\text{if} \quad E\frac\xi{(r+\xi)^2}\ge 0\cr
&\pi\quad&&\text{if} \quad E\frac\xi{(r-\xi)^2}\le 0\cr
&\aligned
&\text{the unique solution of the equation}\cr
&E\frac\xi{r^2+\xi^2+2r\xi\cos\f}=0\cr
&(\text{in the variable }\f,\; 0\le \f\le \pi)
\endaligned
\quad&&\text{if} \quad E\frac\xi{(r+\xi)^2} <0<
E\frac\xi{(r-\xi)^2}.
\endalignedat \right. \tag2.4
$$
The relation
$$
\left.\frac{\partial H(r,\f)}{\partial \f}\right|_{\f=\fb}=0  \tag2.5
$$
holds.}\medskip
 
Define the function $E(r,\f)=r\dfrac\partial{\partial r}H(r,\f)$ and
$G(r)=E(r,\fb(r))$.
\medskip \noindent {\bf Lemma 2.} {\it $G(r)$ is a continuous and
strictly monotone increasing function.}
\medskip Before the proof of Lemma 2 we prove the following technical
Lemma~A. \medskip\noindent
{\bf Lemma A.} {\it The function $H(z)$ defined in formula (1.3) is
analytic in the set $\bold C\setminus(-\bold\Sigma)$ and the
functions $K^\pm(z)$, the analytical continuation of the functions
defined in formula (2.1), are analytic in the set $\bold
C\setminus (\mp\bold\Sigma)$, where $\bold C$ is the space of complex
numbers, and $\bold\Sigma$ is the support of the distribution of the
random variable $\xi$. In particular, $K^\pm(r)$ is continuous in the
points $r\in\Cal A^\pm$. The numbers~$r$ satisfying the equation
$E\dfrac\xi{(\xi\pm r)^2}=0$ have no strictly positive
condensation points.}\medskip\noindent
{\bf Lemma 3.} {\it
$$
\aligned
\lim_{r\to\infty} G(r)&=1\cr
\lim_{r\to 0}G(r)&=P(\xi=0)\quad (=0\text{ if the distribution of $\xi$
has no atom in 0.)}
\endaligned \tag2.6
$$
The second derivative of $H(r,\f)$ with respect to the variable $\f$ is
non-positive in the point $\fb(r)$, and it can be zero only if either
$E\dfrac{\xi}{(r+\xi)^2}=0$ (in which case $\fb(r)=0$) or if
$E\dfrac{\xi}{(r-\xi)^2}=0$ (in which case $\fb(r)=\pi$). More
explicitly,
$$
\alignedat 2
\left.\frac{\partial^2 }{\partial \f^2}H(r,\f)\right|_{\f=\fb(r)}&=
-2E\frac{r^2\xi^2\sin^2\f}{(r^2+\xi^2+2r\xi\cos\f)^2}\quad&&\text{if
}0<\fb(r)<\pi
\cr \left.\frac{\partial^2 }{\partial \f^2}H(r,\f)\right|_{\f=\fb(r)}&=
-E\frac{r\xi}{(r+\xi)^2}\;\;(=-K^+(r))\quad &&\text{if }\fb(r)=0\cr
\left.\frac{\partial^2 }{\partial \f^2}H(r,\f)\right|_{\f=\fb(r)}&=
E\frac{r\xi}{(r-\xi)^2}\;\;(=-K^-(r))\quad &&\text{if }\fb(r)=\pi.
\endalignedat \tag2.7
$$
}\medskip
The above relations imply that the saddle point equation (1.4) (together
with property (1.5)) has a unique solution for $P(\xi=0)<\alpha(n)<1$,
since a pair $(r,\f)$ is a solution if and only if
$\f=\fb(r)$, where $\fb(r)$ is defined in Lemma~1, and $G(r)=\alpha(n)$.
 
Let us rewrite formula (1.2) in the form
$$
S^{(k)}(n)=\Re\( \frac 1{\pi}\int_{0}^{\pi}\exp\left\{ Z_n(r,\f)
\right\}\,d\varphi \) \tag2.8
$$
with
$$
Z_n(r,\f)=\sum_{j=1}^n\beta_j(r,\f) \tag2.9
$$
and
$$
\aligned
\beta_j(r,\f)&=\frac12\log\(r^2+\xi^2_j+2r\xi_j\cos\f\)\\
&\qquad+i\arccos\frac {r\cos\f+\xi_j}
{(r^2+\xi_j^2+2r\xi_j\cos\f)^{1/2}}- \alpha(n)(\log r+i\f),
\endaligned
\tag2.10
$$
where $r$ is the first coordinate of the solution $(r,\fb)$ of the fixed
point equation (1.4) and (1.5). We shall give a good approximation of
$S^{(k)}(n)$ in Section~4. To get it we impose
the following \medskip \noindent
{\bf Property B.} {\it Let $(r,\fb)=(r(\alpha^*),\fb(r(\alpha^*))$ be
the solution of the fixed point equation (1.4) (together with relation
(1.5)), if $\alpha(n)$ is replaced by
$\alpha^*=\lim\limits_{n\to\infty} \alpha(n)$. Then
$$
E\frac \xi{(r\pm \xi)^2}\neq0 \quad\text{for }r=r(\alpha^*).
$$
}\medskip
The integral in formula (2.8) can be well estimated. To do this we
apply a Taylor expansion for $\beta_j(r,\f)$ in the variable
$\f$ around the saddle point $\fb$ and then sum it up to get a good
estimate for $Z_n(r,\f)$ defined in (2.9). The coefficients of this
Taylor expansion are random. But since the random functions
$\beta_j(r,\fb)$ are independent, their sum can be well approximated,
because of the laws of large numbers, by their expected values
multiplied with $n$. The expected value of the first Taylor
coefficient is zero because of (1.4). Indeed, the real part equals
$\dfrac{\partial H(r,\f)}{\partial \f}=0$, and the imaginary part equals
$$
\dfrac{\partial}{\partial \f}E\arccos\frac{r\cos
\f+\xi}{(r^2+\xi^2+2r\xi\cos\f)^{1/2}}-\alpha(n)=
r\dfrac{\partial H(r,\f)}{\partial r}-\alpha(n)=0       \tag2.11
$$
in the point of solution $(r,\fb)$ of (1.4). The identity (2.11) can be
obtained by standard calculation. But it is worth mentioning that this
identity has a deeper reason. There are identities between the partial
derivatives of the real part and analytic part of a complex analytic
function, and the identity (2.11) expresses such properties formulated
in polar coordinate system.
 
By Lemma~3 the expected value of the second partial derivative of the
real part of $\beta_j(r,\f)$ with respect to the variable $\f$ is
non-positive in the asymptotic saddle point $(r,\fb(r))$, and it is
strictly negative if Property~B holds. In this case the integral (2.8)
is essentially concentrated in a small neighbourhood of the point
$\fb(r))$ with probability almost one (depending on $n$). In
this small neighbourhood of the point $\f(r)$ a small error
is committed if all terms $\beta_j(r,\f)$ in (2.8) are replaced by their
Taylor expansion around the point $\fb$ up to the second
term. In such a way the integral in (2.8) can be approximated by a
Gaussian integral which can be explicitly calculated. The above
indicated calculation will be worked out in Section~4. Some additional
technical difficulties arise if we want to show that the error term
obtained in this calculation is negligible also if the real part of the
integral in formula (2.8) is considered. To prove this fact we have
to know that the integral in (2.8) with probability almost one is such a
complex number whose angle with the imaginary axis is not too small. We
can prove this only under some additional restriction formulated a bit
later. We introduce a condition which we shall call the
stability of the level $\alpha^*=\lim\limits_{n\to\infty} \alpha(n)$.
In Proposition~B of Section~5 we prove a limit theorem which helps us to
overcome the above difficulties if the above mentioned stability
condition holds. The proofs in Section~5 are independent of the rest of
the paper. The arguments formulated above lead to a result formulated
in Lemma~4.
 
Before its formulation let us remark that by the last statement of
Lemma~A Property~B is not a strong restriction. The exceptional set of
the numbers $\alpha^*$ where it does not hold has no condensation
points in the open interval $(P(\xi=0),1)$. Moreover, in certain cases
we know that this set is empty. This is the case for instance if $\xi$
has a symmetric distribution, since under this condition $\fb(r)=\pi/2$
for all $r>0$. If Property~B does not hold, then a more complicated
picture arises. In this case not only the first but also the second
derivative  of the function $H(r,\f)-\alpha\log r$ disappears in the
saddle point. Hence a more sophisticated method has to be applied and
only weaker results can be obtained in this case. We shall not discuss
this question in the present paper.
\medskip\noindent{\bf Lemma 4.} {\it Let the distribution of $\xi$
satisfy Property~A and (2.3). Beside this, let property~B be satisfied
with $r^*=\lim\limits_{n\to\infty}r_n$, where $r_n$ is the solution of
the asymptotic saddle point equation (1.4) (together with (1.5)) with
the parameter $\alpha(n)$. Let us also assume that the level
$\alpha^*=\lim\limits_{n\to\infty}\alpha(n)$ is stable. (This notion
will be introduced a bit later.) Put
$$
\bar S^{(k)}(n)=\left\{ \aligned
&\frac{\sqrt2}{\sqrt{Kn\pi}}\exp\left\{nA_0+\sqrt
nS_0-U_1\right\}\cos\(nB_0+T_0-U_2-\frac\omega2\) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\text{if } 0<\fb(r^*)<\pi  \\
&=\frac{1}{\sqrt{2|A_2|\pi n}}\exp\left\{\frac{T_1^2}{2A_2}
+nA_0+\sqrt n S_0\right\}\quad\text{if }\fb(r^*)=0\\
&=(-1)^{k(n)}\frac{1}{\sqrt{2|A_2|\pi n}}\exp\left\{\frac{T_1^2}
{2A_2} +nA_0+\sqrt n S_0\right\}\quad\text{if }\fb(r^*)=\pi.
\endaligned\right. \tag2.12
$$
where the random variables $S_0=S_0(n)$, $T_0=T_0(n)$, $S_1=S_1(n)$,
$T_1=T_1(n)$ which are sums of independent random variables are defined
in (4.8), (4.9) and (4.1), (4.2), the random variables $U_1=U_1(n)$,
$U_2=U_2(n)$ which are their transforms in (4.14). The constants
$A_0=A_0(n)$, $B_0=B_0(n)$, $A_2=A_2(n)$, $K$ and $\omega=\omega(n)$
are defined in (4.3), (4.4) and $(4.14')$. Then
$$
\frac{S^{(k)}(n)}{\bar S^{(k)}(n)} \Rightarrow 1
$$
where $\Rightarrow$ denotes convergence in probability.
}\medskip
Lemma 4 plays a crucial role in our investigation, because it enables
us to replace the expression $S^{(k)}(n)$ introduced in $(1.1)$  by
$\bar S^{(k)}(n)$ defined in (2.12) when we are interested in its limit
behaviour. The expression $\bar S^{(k)}(n)$ is a functional of the
random variables $S_0(n)$, $S_1(n)$, $T_0(n)$ and $T_1(n)$ which are
normalized sums of independent random variables. The asymptotic
behaviour of $S_0(n)$, $S_1(n)$ and $T_1(n)$ is described by the
central limit theorem while that of $T_0(n)$ by limit theorems for
sums of independent random variables on the compact group $[0,1]$ mod~1,
where the group action is summation modulo 1. But these classical
results are not sufficient for our purposes, we also want to control
the limit of the joint distribution of the above random variables. Hence
we formulate the following Proposition~A whose proof will be given in
Section~5. It implies that $T_0(n)$ is asymptotically independent from
the other partial sums, because it takes values on a compact group,
while the other partial sums on a non-compact group. Before formulating
this result we introduce some notations and make some remarks.
 
We shall identify the group $G=[0,1)$ with summation modulo 1 with the
unit circle. Let us remark that the closed subgroups $G_0$ of $G$ are
the group $G$ itself and the discrete groups of the form
$G_0=\left\{\dfrac jp;\;j=0,\dots,p-1\right\}$ with some positive
integer $p$. A coset of a finite subgroup $G_0$ is of the form
$G_0+\alpha$ with some $0\le\alpha<1$. For all probability measures
$\mu$ on $(0,1]$ there is a smallest closed set, called the support of
the measure, whose $\mu$ measure is one. For all probability measures
$\mu$ there is a minimal coset $G_0+\alpha$ which contains the support
of $\mu$. This means that the $\mu$ measure of this coset is 1, and all
cosets with this property contain this coset. If no coset of a finite
subgroup of $G$ has this property, then we call the whole group $G$
the minimal coset which contains the support of the measure $\mu$.
Now we formulate the following \medskip \noindent
{\bf Proposition A.} {\it Let $(X_n,Y_n)$, $n=1,2,\dots$, be a
sequence of i.i.d. random vectors such that $X_n$ is a random vector in
$R^k$ with expectation zero and covariance matrix $\Sigma$,  $Y_n$ is a
random variable on the unit circle $G=[0,1)$. Let $G_0+\alpha$ be the
minimal coset which contains the support of the distribution of $Y_n$.
Put $U_n=\dfrac1{\sqrt n}\sum\limits_{s=1}^n X_s$,
$V_n=\sum\limits_{s=1}^nY_s-n\alpha$. Then the joint distribution of
$(U_n,V_n)$ tends to the distribution of a random vector $(U,V)$,
where $U$ has normal distribution with expectation zero and covariance
$\Sigma$, $V$ is uniformly distributed on the subgroup $G_0$ of $G$, and
the random variables $U$ and $V$ are independent. In the case $G_0=G$,
$\alpha$ can be chosen in an arbitrary way, e.g.\ $\alpha=0$.}
\medskip
The result of Proposition~A is not sufficient in itself for our
purposes. The reason for this is that the distributions of the random
variables we are investigating depend on a parameter $\alpha(n)$. This
parameter satisfies the relation $\alpha(n)\to\alpha^*$, but it may
depend on $n$. Hence we need such a version of Proposition~A
where the distribution of the random variables $X_j=X_j{(n)}$ and
$Y_j=Y_j{(n)}$, $j=1,\dots,n$, may weakly depend on $n$. Let us remark
that in the limit theorems for sums of independent random variables on
a compact group $G$ no normalization is taken, hence even a small
perturbation of the summands may radically change the limit
distribution of their sums. Nevertheless, we show that in the case when
the distribution of $Y_n$ is close to a measure which is not
concentrated in a coset of a closed finite subgroup a version of
Proposition~A can be proved where the distribution of the summands may
depend on $n$. To formulate this result first we introduce the following
definition.
\medskip\noindent{\bf Definition.} {\it We call a probability measure
$\mu$ on the group $G=[0,1)$,~{\rm mod~1} stable if for all finite
cosets $K=\left\{\dfrac jp+c,\;j=0,\dots,p-1\right\}$, with a positive
integer $p$ and $0\le c<1$ \ $\mu(K)<1$, or in other words, the minimal
coset which contains the support of the measure $\mu$ is the whole
group $G$.} \medskip
This terminology for stable distribution differs from the traditional
one, but since we apply it on a different space, hopefully it causes no
confusion. Now we formulate the following result. \medskip\noindent
{\bf Proposition B.} {\it For all $n$ let $(X_j{(n)},Y_j{(n)})$,
$j=1,2,\dots,n$, be a sequence of i.i.d.\ random vectors with the
following properties:  $X_j{(n)}$ are i.i.d.\ random vectors in $R^k$,
$EX_j(n)=0$, the relation $E\|X_1(n)-X\|^2\to0$ holds with a random
variable $X$ in $R^k$, $EX=0$, which has a covariance matrix $\Sigma$,
$Y_j{(n)}$ is a random variable on the unit circle $[0,1)$ with a
distribution $\mu_n$ on $[0,1)$ such that $\mu_n\Rightarrow \mu$, and
$\mu$ is a stable probability measure on $[0,1)$, where $\Rightarrow$
denotes weak convergence of measures. Define the random variables
$U_n=\dfrac1{\sqrt n}\sum\limits_{s=1}^n X_s{(n)}$ and
$V_n=\sum\limits_{s=1}^nY_s{(n)} \mod \;1$. Then the joint
distribution of $(U_n,V_n)$ tends to the distribution of a random
vector $(U,V)$, where $U$ has normal distribution with expectation
zero and covariance $\Sigma$, $V$ is uniformly distributed on
$G=[0,1)$, and the random variables $U$ and $V$ are independent.}
\medskip
Propositions A and B hold because one of the coordinates of the random
vectors we are summing up take value in a compact while the other
component in a non-compact group. Results similar to Proposition~A
can be found in [6] in a more general setting, but to find the right
generalization of Proposition~B seems to be an interesting open
question.
 
The above results enable us to investigate the limit behaviour of the
random variable $S^{(k)}(n)$ defined in (1.1). But because of the
conditions we had to impose in the limit theorem formulated in
Proposition~B we can prove these results only under certain
restrictions. Let us introduce the following terminology:
\medskip\noindent
{\bf Definition.} {\it We call the level $\alpha^*$ stable if one of the
following conditions are satisfied.\medskip
\item{1.)} Either $E\dfrac{\pm\xi}{(r\pm\xi)^2}>0$ for
$r=r(\alpha^*)$, i.e.\ either $\fb(\alpha)=0$ or $\fb(\alpha)=\pi$
if $\alpha$ is in a small neighbourhood of $\alpha^*$.
\item{2.)} or $0<\fb(\alpha^*)<\pi$, and the distribution of the random
variable
$$
Y=\frac1{2\pi}\arccos\dfrac{r\cos\fb+\xi}
{(r^2+\xi^2+2r\xi\cos\fb)^{1/2}},
$$
where $r=r(\alpha^*)$ and $\fb=\fb(\alpha^*)$
is a stable distribution on the unit circle $[0,1)$.}  \medskip
We can give a good asymptotic of the symmetric statistics
$S^{(k)}(n)$ if $\dfrac{n-k}n=\alpha(n)\to\alpha^*$ with a stable level
$\alpha^*$.
 
If $0<\fb(\alpha^*)<\pi$, then the second condition of the stability of
$\alpha^*$ holds in the generic case, but the description of the
exceptional numbers $\alpha^*$ and distributions $F$ seems to be a
hard number theoretic problem. Now we formulate the following Theorem.
\medskip\noindent
{\bf Theorem 1.} {\it Let Property~A and relation (2.3) hold, and let
$\alpha^*$ be a stable level. If $\alpha(n)=\dfrac{n-k}n\to\alpha^*$ as
$n\to\infty$, then the random variables
$$
\frac{\log |S^{(k)}(n)|-nA_0(n)}{\sqrt n} \tag2.13
$$
(with $S^{(k)}(n)$ defined in (1.1)) converge in distribution to the
normal law with expectation zero and variance $\text{Var}\,\eta$, where
$\eta=\eta(\fb)=\dfrac12\log\(r(\alpha^*)^2+\xi^2+2r(\alpha^*)
\xi\cos\fb(\alpha^*)\)$, $(r(\alpha^*),\f(\alpha^*))$ is the solution of
the saddle-point equation (1.4) if the number $\alpha(n)$ is replaced by
$\alpha^*=\lim\limits_{n\to\infty}\alpha(n)$,
and $A_0=A_0(n)$ is defined in (4.3). $|S^{(k)}(n)|$ can be replaced by
$S^{(k)}(n)$ in the case $\fb(\alpha^*)=0$, by $(-1)^kS^{(k)}(n)$ in
the case $\fb(\alpha^*)=\pi$ in (2.13), while in the case
$0<\fb(\alpha^*)<\pi$ $P(\,\text{\rm sign}\,S^{(k)}(n)\to1)=1/2$ and
$\log |S^{(k)}(n)|$ and $\text{\rm sign}\, S^{(k)}(n)$ are
asymptotically independent.} \medskip\noindent
Theorem 1 does not contain the result of [4], where limit theorem is
given for a normalized version of $S^{(k)}(n)$ (without logarithm) if
the random variables $\xi_j$ have the distribution
$P(\xi_j=1)=P(\xi_j=-1)=1/2$. In this case
the random variable $\eta$ is constant, Var$\,\eta=0$, and the limit
(2.13) is degenerate. In the following Lemma~5 we describe those
distributions $F$ and levels $\alpha^*$ for which the limit distribution
in Theorem~1 is degenerate. Then we shall describe the limit behaviour
of $S_n^{(k)}$ in such cases. \medskip\noindent
{\bf Lemma 5.} {\it The random variable
$\eta=\eta(\alpha^*)=\frac12\log\(r(\alpha^*)^2+\xi^2+2r\xi
\cos\fb(\alpha^*)\)$ appearing in Theorem~1 is constant, if an $F$
distributed random variable $\xi$ is concentrated in two points, i.e.\
there are two numbers $x_1$, $x_2$ such that $P(\xi=x_1)=p$,
$P(\xi=x_2)=q=1-p$, and one of the following  conditions is satisfied.
\item{a.)} $0<\fb(\alpha^*)<\pi$, in which case $E\xi=px_1+qx_2=0$,
$\alpha^*>1-4pq$.
\item{b.)} $\fb(\alpha^*)=0$, in which case
$\alpha^*=-\dfrac{(p-q)(x_1+x_2)}{x_1-x_2}$, $E\xi=px_1+qx_2\ge0$ and
$x_1+x_2<0$.
\item{c.)} $\fb(\alpha^*)=\pi$, in which case
$\alpha^*=-\dfrac{(p-q)(x_1+x_2)}{x_1-x_2}$, $E\xi=px_1+qx_2\le0$,
and $x_1+x_2>0$.} \medskip
In Theorem 2 we describe the limit behaviour of $S^{(k)}(n)$ in case a.)
of Lemma~5. It contains the result of [4]. \medskip\noindent
{\bf Theorem 2.} {\it Let the distribution of the random variable $\xi$
have the form $P(\xi=x_1)=p$, $P(\xi=x_2)=q=1-p$, $px_1+qx_2=0$, i.e.\
$E\xi=0$. Let $\frac{n-k}n=\alpha(n)\to \alpha^*$ with some stable
level $\alpha^*$ such that $1>\alpha^*>1-4pq$. Then the random variables
$$
\frac{\sqrt{K\pi n}}{\sqrt2}e^{- A_0(n)}S^{(k)}(n)
$$
converge in distribution to the random variable
$\exp\left\{\dfrac{A_2(S^2-T^2)+2B_2ST}{2(A_2^2+B_2^2)}\right\} \cos Z$
as $n\to\infty$, where the constants $A_0$, $A_2$, $B_2$ and $K$ are
defined in formulas (4.3), (4.4), $(4.14')$, more precisely they are the
limits of these quantities depending on $n$ as $n\to\infty$,
$(S,T)$ is a Gaussian random vector with expectation zero, $Z$ is a
random variable, uniformly distributed in $[0,2\pi)$ and independent of
the vector $(S,T)$, and
$$
\aligned
&ES^2=\text{\rm Var}\,\frac{-r\xi\sin\fb}
{r^2+\xi^2+2r\xi\cos\fb}\;\qquad ET^2=\text{\rm Var}\,
\frac{r\xi\cos\fb+r^2}{r^2+\xi^2+2r\xi\cos\fb},\\
&\text{\rm Cov}\,(S,T)=\text{\rm Cov}\,\(\frac{-r\xi\sin\fb}
{r^2+\xi^2+2r\xi\cos\fb},\frac{r\xi\cos\fb+r^2}{r^2+\xi^2+2r\xi\cos\fb}\),
\endaligned \tag2.14
$$
where $r=r(\alpha^*)$, $\fb=\fb(\alpha^*)$.} \medskip
Finally, in Theorem $2'$ we describe the limit behaviour of $S^{(k)}(n)$
in the case when the conditions of Part b.) of Lemma~5 hold. The case
when the conditions of Part c.) hold can be obtained by applying this
result for the random variables $-\xi_j$ which  satisfy Part~b.).
\medskip\noindent
{\bf Theorem 2$\bold '$.} {\it Let the distribution of $\xi$ satisfy the
following conditions: $P(\xi=x_1)=p$, $P(\xi=x_2)=q=1-p$ with some
$x_1$, $x_2$ and $p$ such that $px_1+qx_2>0$ and $x_1+x_2<0$,
$x_1>x_2$. Put $\alpha^*=\dfrac{(p-q)(-x_1-x_2)}{x_1-x_2}$. If $\dfrac
{n-k}n=\alpha(n)\to\alpha^*$, then the symmetric polynomial
$S^{(k)}(n)$ satisfies the following limit theorem:
$$
\align
&\sqrt{2|A_2|\pi n} e^{-nA_0(n)}
S^{(k)}(n)\Rightarrow \exp\left\{\frac {T^2}{A_2}\right\}\quad
\text{if } \sqrt n(\alpha(n)-\alpha^*)\to 0 \\
&{\sqrt{2|A_2|\pi n}} e^{-nA_0(n)}
S^{(k)}(n)\Rightarrow \exp\left\{\frac {T^2}{A_2}+cLV\right\}\quad
\text{if } \sqrt n(\alpha(n)-\alpha^*)\to c, \;\, 0<|c|<\infty,
\endalign
$$
where $L=\dfrac{\sqrt{pq}(x_1-x_2)}{px_1+qx_2}$,
$T=-\dfrac{x_1+x_2}{x_1-x_2}V$, and $V$ is a standard normal
random variable.}\medskip
If $|\sqrt n(\alpha(n)-\alpha^*)|\to \infty$, there is not such a
natural scaling of $S^{(k)}(n)$ as in the previous cases.
 
\beginsection 3. The solution of the fixed point equation.
 
In this Section we prove Lemmas 1, 2 and 3 which imply that there is a
unique solution of equation (1.4), $0\le \f\le \pi$, which also
satisfies relation (1.5).
\medskip\noindent {\it Proof of Lemma 1.}\/ Let us define the
function $L(r,\psi)=\dfrac12 E\log(r^2+\xi^2+2r\xi\psi)$, $-1\le
\psi\le 1$. This function is obtained if $\psi$ is written
instead of $\cos\f$  in the function $H(r,\f)$. It is a concave
function of the variable $\psi$ in the open interval $-1<\psi<1$ for
all $r>0$, since its second derivative is negative. The behaviour of
the function $L(r,\psi)$ in the end point $\psi=1$ can be investigated
by means of the following observation. There is a sufficiently small
$\e>0$ such that in the interval $1-\e<\psi<1$ either $L(r,\psi)$ is
monotone decreasing and the derivative $\dfrac{\partial
L(r,\psi)}{\partial\psi}$ is negative or $L(r,\psi)$ is monotone
increasing and the derivative $\dfrac{\partial
L(r,\psi)}{\partial\psi}$ is positive. In the first case
$$
L(r,1)-L(r,\psi)=\frac12E\log\(1+\frac{2(1-\psi)r\xi}{r^2+\xi^2+2r\xi\psi}\)\le
E\frac{(1-\psi)r\xi}{r^2+\xi^2+2r\xi\psi}=(1-\psi)\frac{\partial
L}{\partial\psi}<0,
$$
and $L(r,1)<\sup L(r,\psi)$.
 
In the second case it follows from formula (2.3) and Fatou's lemma that
$$
0\le\limsup_{\psi\to1}\frac{\partial L}{\partial\psi}=\limsup_{\psi\to1}
E\frac{r\xi}{r^2+\xi^2+2r\xi\psi}\le E\frac{r\xi}{(r+\xi)^2}=rK^+(r),
$$
where the function $K^+(r)$ is defined in (2.1). Hence $r\in \Cal A^+$,
and Property~A can be applied. This implies in particular that
$L(r,1)=\lim\limits_{\psi\to 1} L(r,\psi)=
\sup\limits_{0\le \psi\le 1} L(r,\psi)$. Similarly, $\psi=-1$ is the
maximum of $L(r,\psi)$ if and only if the function $L(r,\psi)$ is
monotone decreasing in the interval $(-1,-1+\e)$ with a sufficiently
small $\e>0$, and $r\in \Cal A^-$, i.e. $K^-(r)\ge0$. In particular, the
function $L(r,\psi)$ is continuous in the point $\psi=-1$ in this case.
 
The above results imply that the function $H(r,\f)$ has a unique maximum
in the interval $0\le \f\le \pi$. The maximum is in the point $\f=0$ if
the function $L(r,\psi)$ has its maximum at $\psi=1$ which holds if
$K^+(r)\ge 0$. It has its maximum at $\f=\pi$ if $L(r,\psi)$ has its
maximum at $\psi=-1$ and $K^-(r)\ge 0$. These statements are equivalent
to the first two lines of formula (2.4). The maximum is in the open
interval $0<\f<\pi$ if $-K^-(r)<0<K^+(r)$. In this case
$\dfrac{\partial H(r,\f)}{\partial \f}=0$ in the  place of
maximum, and since the order of differentiation and expectation can be
changed, this fact implies the third line of formula (2.4). Finally,
relation (2.5) also holds for $\fb=0$ and $\fb=\pi$. To see this,
observe that since $r\in \Cal A^-$ if $\fb=0$, $r\in \Cal A^+$ if
$\fb=\pi$, $\dfrac{\partial H(r,\f)}{\partial \f}=0$ in the place of
maximum $\fb$, and the order of differentiation and expectation can be
changed in this case too. \medskip
Let us introduce the notation $U=U(r,\xi,\f)=r^2+\xi^2+2r\xi\cos\f$. Now
we turn to the \medskip\noindent
{\it Proof of Lemma A.}\/ If $z_0=r_0e^{i\varphi_0}\notin-\bold\Sigma$,
and $\xi\in\bold\Sigma$ then for all $z=re^{i\varphi}$ in a sufficiently
small neighbourhood of $z_0$ the number $|z+\xi|^2=U(r,\xi,\f)\ge C>0$
with an appropriate number $C=C(z_0)$. Hence the function $\log
U(r,\xi,\f)$ is analytic in such a small neighbourhood of $z_0$, and it
is separated from $-\infty$ (independently of $\xi\in\bold\Sigma$).
Then, since $\log U(r,\xi,\f)\le\const(|\xi|+r)$, and relation (2.3)
holds, we get by taking expectation that $H(z)=\frac12E\log
U(r,\xi,\f)$ is analytic in a small neighbourhood of~$z_0$.
 
Similarly, if $z_0\notin\-\mp\bold\Sigma$, $\xi\in\bold\Sigma$ and $z$
is in a small neigbourhood of $z_0$, then $\left|\dfrac\xi{(\xi\pm
z)^2}\right|\le C<\infty$, and taking expectation we get that the
functions $K^\pm(z)$ are analytic in the domain $\bold C\setminus
(\mp\bold\Sigma)$. In particular, Property~A implies that the function
$K^\pm(r)$ is continuous in the points $r\in \Cal A^\pm$.
 
Moreover, the function $K^\pm(r)$ defined for all $r>0$  is
upper semicontinuous, hence the sets $\Cal A^\pm$ defined in (2.2) are
closed subsets of the positive numbers. We show that there is no
sequence $r_n$, $n=1,2,\dots$, with a limit
$0<r=\lim\limits_{n\to\infty} r_n<\infty$ such that
$K^\pm(r_n)=0$ for all $n$. Indeed, the limit $r$ would be also in the
set $\Cal A^\pm$, and because of Property~A the relation
$d(r,\mp\bold\Sigma)>0$ would hold. This would imply that
$K^\pm(z)\equiv0$ in the domain of analiticity of the function
$K^\pm(z)$. This relation also would imply that $E\dfrac\xi{\xi\pm
z}=0$ on the set $\Im z>0$, since the derivative of this function is
$K^\pm(z)\equiv0$, and as a consequence it is a constant function. Then
choosing $z=iu$, $u\to\infty$ we get that this constant is zero. On the
other hand, we get with the choice $z=iu$, $u\to0$ that this constant
is $P(\xi\neq0)\neq0$, and this is a contradiction. \medskip
Now we turn to the \medskip\noindent
{\it Proof of Lemma 2.}\/ We shall prove that
$$
\frac {dG(r)}{dr}>0\quad\text{if } E\frac\xi{(r+\xi)^2}
<0<E\frac\xi{(r-\xi)^2}\quad\text{(or equivalently, if
$0<\fb(r)<\pi$),} \tag3.1
$$
and also
$$
\frac {dG(r)}{dr}>0\qquad\text{if}\quad
E\frac\xi{(r+\xi)^2}>0\quad
\text{or}\quad E\frac\xi{(r-\xi)^2}<0. \tag3.2
$$
Finally we show that the function $G(r)$ is continuous for all $r>0$.
This continuity, the last statement of Lemma~A,
together with formulas (3.1)  and (3.2) imply that in an interval
$[a,b]$, $0<a<b<\infty$,  $\dfrac{G(r)}{dr}>0$  with the possible
exception only of finitely many points. Lemma~2 follows from this fact.
 
To prove relation (3.1) observe that in this case
$E\dfrac\xi{r^2+\xi^2+2r\xi\cos\fb(r)}=0$. This identity determines the
function $\fb(r)$ in the small neighbourhood of a point $(r,\fb(r))$.
The implicit function theorem enables us to calculate the function
$\fb'(r)$. We get that
$$
\fb'(r)=\frac{E\dfrac{2\xi(r+\xi\cos\fb)}{U^2}}
{E\dfrac{2\xi^2r\sin\fb}{U^2}}
=\frac{\cos\fb}{r\sin\fb}+\frac{E\dfrac \xi{U^2}}{\sin\fb
E\dfrac{\xi^2}{U^2}}. \tag3.3
$$
Exploiting again that the third line of formula (2.4) holds in this
case, we get that
$$
G(r)=E(r,\fb(r))=E\frac{r^2+r\xi\cos \fb(r)}{U(r,\xi,\fb(r))}
=E\frac{r^2}{U(r,\xi,\fb(r))}=1-E\frac{\xi^2}{U(r,\xi,\fb(r))} \tag3.4
$$
and
$$
\aligned
\frac
{dG(r)}{dr}&=E\frac{2\xi^2(r+\xi\cos\fb(r))}{U^2(r,\xi,\fb(r))}-\fb'(r)
E\frac{2r\xi^3\sin\fb(r)}{U^2(r,\xi,\fb(r))}\\
&=E\frac{2r\xi^2}{U^2(r,\xi,\fb(r))}-\frac{E\frac\xi{U^2(r,\xi,\fb(r))}
E\frac{2r\xi^3}{U^2(r,\xi,\fb(r))}}{E\frac{\xi^2}{U^2(r,\xi,\fb(r))}},
\endaligned
$$
if $0<\fb(r)<\pi$. Hence relation (3.1) is equivalent to the inequality
$$
E\frac{r^2 \xi}{U^2} E\frac{\xi^3}{U^2}<\(E\frac{r\xi^2}{U^2}\)^2,
$$
or since the third line in formula (2.4) implies that
$$
E\frac{r\xi^2}{U^2}=\frac1{2\cos\fb}E\frac{2r\xi^2\cos\fb-\xi
U}{U^2}=-\frac1{2\cos\fb}\(E\frac{\xi^3}{U^2}+E\frac{r^2\xi}{U^2}\)
$$
it is also equivalent to the inequality
$$
(4\cos^2\fb-2)E\frac{\xi^3}{U^2}\,E\frac{r^2\xi}{U^2}<
\(E\frac{\xi^3}{U^2}\)^2+\(E\frac{r^2\xi}{U^2}\)^2.
$$
 
The Cauchy--Schwarz inequality implies that the last inequality and
hence relation (3.1) holds. To see that this formula holds with a
strict inequality it is enough to observe that $|4\cos^2\fb-2|<2$ for
$0<\fb<\pi$, and the equations $E\dfrac{\xi^3}{U^2}=0$ and
$E\dfrac{r^2\xi}{U^2}=0$ cannot hold simultaneously. Indeed, they would
imply together with the third line of formula (1.4) for $r>0$ and
$0<\fb<\pi$ that $E\dfrac{\xi^2}{U^2}=0$, and this is impossible.
 
To prove relation (3.2) let us observe that if
$E\dfrac\xi{(r+\xi)^2}>0$, then $\fb(r)=0$, and because of Property~A
the order of differentiation with respect to the variable $r$ and
expectation can be changed when $G(r)$ and $\dfrac{dG(r)}{dr}$ are
calculated. Simple calculation shows that
$G(r)=E(r,\fb(r))=E\dfrac{r}{\xi+r}$,
$\dfrac{dG(r)}{dr}=E\dfrac{\xi}{(r+\xi)^2}>0$, and if
$E\dfrac\xi{(r-\xi)^2}<0$, then $\fb(r)=\pi$, and $\dfrac{dG(r)}{dr}=
-E\dfrac{\xi}{(r-\xi)^2}>0$. These formulas imply (3.2).
 
The above arguments also show the continuity of the function $G(r)$
except the points $r$ such that $E\dfrac\xi{(\xi\pm r)^2}=0$. To prove
the continuity in these points it is enough to show that the function
$\bar\f(r)$ defined in Lemma~1 is continuous in these points. To prove
this observe that in these points either $\bar\f(r)=0$ or
$\bar\f(r)=\pi$. If $\bar\f(r)=0$, then, as we showed in the proof of
Lemma~1, the expression in the third line of formula (2.4) is strictly
negative for this $r$ and $0<\f\le\pi$. This function is uniformly
continuous (analytic) and separated from zero in a small neighbourhood
of the set $\{z\:z=re^{i\f}\}$, with this $r$ and $\e\le\f\le\pi$ for
arbitrary $\e>0$. This implies that $\bar\f(r)$ is continuous in this
exceptional set if $\bar\f=0$. The case $\bar\f=\pi$ can be handled
similarly. Lemma~2 is proved.
\medskip\noindent
{\it Proof of Lemma 3.}\/ Since $G(r)$ is a monotone increasing
function it is enough to prove the formulas in relation (2.6) for a
special sequence $r_n\to\infty$ and $r_n\to0$. To prove the first
relation let us first consider the case when there is a sequence of
numbers $r_n\to\infty$ such that $0<\fb(r_n)<\pi$. By relation (3.4),
Fatou's lemma and the observation $\dfrac{r_n^2}U\to1$,
$\dfrac{\xi^2}U\to0$, $\dfrac{\xi^2}U\ge0$, $\dfrac{r^2}U\ge0$ imply
that
$$
\align
\liminf_{r\to\infty}G(r)&=\liminf_{r\to\infty}E\frac{r^2}U\ge 1,\\
\limsup_{r\to\infty}G(r)&=1-\liminf_{r\to\infty}E\frac{\xi^2}U\le 1,
\endalign
$$
hence the first line of relation of (2.6) holds in this case.
Similarly if $r_n\to0$, $0<\fb(r_n)<\pi$, then $\dfrac{r^2}U\to
I(\xi=0)$ and $\dfrac{\xi^2}U\to1-I(\xi=0)$. Then a similar
argument proves the second line of (2.6) in this case.
 
In the remaining cases, we have because of the continuity of the
function $\bar\f(r)$ either
$\bar\f(r)=0$ and $E\dfrac\xi{(\xi+r)^2}\ge0$ or $\bar\f(r)=\pi$ and
$E\dfrac{-\xi}{(\xi-r)^2}\ge0$ for all $r\ge r_0$ with some $r_0>1$ if
the case $r\to\infty$ is considered. We claim that
$-\bold\Sigma\cap \{r\:r>r_0\}$ is empty for $r>r_0$ in the first case,
and $\bold\Sigma\cap \{r\:r>r_0\}$ is empty for $r>r_0$ is empty in the
second case, where $\bold\Sigma$ denotes the support of the
distribution of the $\xi$. Indeed, if this relation did not hold, then
in the first case one could find by a halving procedure a sequence of
intervals $[a_n,b_n]$ such that $b_n>a_n>r_0$, $b_n-a_n=2^{-n}$,
$F(-b_n)-F(-a_n)\ge K2^{-n}$ with some appropriate $K>0$ for all
$n=1,2,\dots$, where $F$ is the distribution function of the random
variable $\xi$. Let $R$ be the intersection of the intervals
$[a_n,b_n]$, $n=1,2,\dots$. Then $R>r_0$, and we claim that
$E\dfrac\xi{(\xi+R)^2}=-\infty$ which is
a contradiction. This equation holds, because for all $n>0$
$$
E\frac\xi{(\xi+R)^2}\le \int_{-b_n}^{-a_n}\frac x{(R+x)^2}F(\,dx)
+E\xi I(\xi\ge0)\le -\const 2^n+\const, \tag3.5
$$
and we get the above relation as $n\to\infty$. The proof in the case
$\bar\f=\pi$ for $r\ge r_0$ is similar.
 
It follows from the above proved statement, the relation
$\lim\limits_{r\to\infty}\dfrac r{r\pm\xi}=1$ with probability one
and Lebesgue convergence theorem that $\lim\limits_{r\to\infty}G(r)
=\lim\limits_{r\to\infty}E\dfrac r{r\pm\xi}=1$ in this case too.
 
The limit behaviour in the case $r\to0$ can be handled similarly. If
there is no sequence $r_n\to0$ such that $0<\bar\f(r_n)<\pi$,
then there is a number $1>r_0>0$ such that either $\bar\f(r)=0$ or
$\bar\f(r)=\pi$ for all $0<r<r_0$. In the first case
$-\bold\Sigma\cap \{r\:0<r<r_0\}=\emptyset$, and in the second case
$\bold\Sigma\cap \{r\:0<r<r_0\}=\emptyset$. This can be proved similarly
to the case $r\to\infty$ with an estimate similar to (3.5) with the
difference that in this case the relation
$E\dfrac\xi{(R+\xi)^2}I(\xi>0)\le E\dfrac\xi{\xi^2}I(\xi>0)\le
E\dfrac{I(\xi\neq0)}{|\xi|}<\infty$ holds.
 
Finally, as $\lim\limits_{r\to0}\dfrac r{r\pm\xi}=I(\xi=0)$, the
Lebesgue dominated convergence theorem implies that
$\lim\limits_{r\to0}G(r)=\lim\limits_{r\to0}E\dfrac
r{r\pm\xi}=EI(\xi=0)$. Relation (2.6) is proved.
 
We have proved that the  saddle point equation (1.4) and (1.5) has a
unique solution if $P(\xi=0)<\alpha(n)<1$. Let us calculate the
second partial derivative of $F(r,\fb)$ with respect of the variable
$\f$ in the saddle point. We get that
$$
\frac{\partial^2 }{\partial
\f^2}H(r,\f)=-E\frac{r\xi\cos\f}U-2E\frac{r^2\xi^2\sin^2\f}{U^2},
$$
in a general point $(r,\f)$. Then a simple substitution implies
formula (2.7). Lemma~3 is proved.
\medskip\noindent
{\bf 4. Asymptotic approximation for the symmetric polynomial
$S^{(k)}(n)$.}
\medskip\noindent
Let us consider the solution $(r_n,\fb_n)$ of the asymptotic saddle
point equation (1.4) which also satisfies relation (1.5). Let us remark
that these numbers depend on $n$ because of the function $\alpha(n)$ at
the right-hand side of formula (1.4). On the other hand, if
$(r(\alpha^*),\fb(\alpha^*))$ denotes the solution of the equation
(1.4) with the modification that the number $\alpha(n)$ is replaced
by $\alpha^*=\lim\limits_{n\to\infty}\alpha(n)$ in it, then
$\lim\limits_{n\to\infty}r_n=r(\alpha^*)$, and
$\lim\limits_{n\to\infty}\fb_n=\fb(\alpha^*)$. Indeed, it
follows from Lemma~2 that $\lim\limits_{n\to\infty}r_n=r(\alpha^*)$,
since the function $G(r)$ which was so defined that the number $r_n$ is
the solution of the equation $G(r)=\alpha(n)$ is a continuous and
strictly monotone function. Then it follows from Lemma~1 that the
relation $\lim\limits_{n\to\infty}\fb_n=\fb(\alpha^*)$ also holds.
 
We want to make a Taylor expansion of the function $\beta_j(r_n,\f)$
defined in formula (2.10) in the variable $\f$ around the point
$(r_n,\fb_n)$. For this end we introduce some notations. Put
$$
\aligned
\eta_j^{(0)}&=\eta_j^{(0)}(n)=\Re\(\beta_j(r_n,\fb_n)-E\beta_j(r_n,\fb_n)\)
=\frac12\log\(r^2_n+\xi^2_j+2r_n\xi_j\cos \fb_n\)\\
&\qquad -\frac12 E\log\(r^2_n+\xi^2+2r_n\xi\cos\fb_n\), \\
\zeta_j^{(0)}&=\zeta_j^{(0)}(n)=\Im\(\beta_j(r_n,\fb_n)-E\beta_j(r_n,\fb_n)\)
=\arccos\frac
{r_n\cos\fb_n+\xi_j}{(r_n^2+\xi_j^2+2r_n\xi_j\cos\fb_n)^{1/2}} \\
&\qquad-E\arccos\frac{r_n\cos\fb_n+\xi}
{(r_n^2+\xi^2+2r_n\xi\cos\fb_n)^{1/2}},
\endaligned  \tag4.1
$$
$$
\align
\eta_j^{(1)}&=\eta_j^{(1)}(n)=\left.\frac{\partial}{\partial \f}
\Re\(\beta_j(r_n,\f)-E\beta_j(r_n,\f)\)\right|_{\f=\fb_n}
=\left.\frac{\partial}{\partial \f}
\Re\beta_j(r_n,\f)\right|_{\f=\fb_n}\\
&=-\frac{r_n\xi_j\sin\fb_n}
{r_n^2+\xi_j^2+2r_n\xi\cos\fb_n}, \tag4.2   \\
\zeta_j^{(1)}&=\zeta_j^{(1)}(n)=\frac{\partial}{\partial
\f}\Im\(\beta_j(r_n,\fb_n)
-E\beta_j(r_n,\fb_n)\) =\left.\frac{\partial}{\partial
\f}\Im\beta_j(r_n,\f)\right|_{\f=\fb_n} \\
&=\frac{r_n\xi_j\cos\fb_n+r^2_n}{r_n^2+\xi_j^2+2r_n\xi_j\cos\fb_n}
-\alpha(n),
\endalign
$$
(in the last identity we applied the same calculation as in formula
(2.11))
$$
\aligned
A_0&=A_0(n)=
E\Re\beta_j(r_n,\fb_n)=\frac12E\log\(r_n^2+\xi^2+2r_n\xi\cos
\fb_n\)-\alpha(n)\log r_n,\\
B_0&=B_0(n)-E\Im\beta_j(r_n,\fb_n)=E\arccos\frac
{r_n\cos\fb_n+\xi}{(r_n^2+\xi^2+2r_n\xi\cos\fb_n)^{1/2}}-\alpha(n)\fb_n,
\endaligned         \tag4.3
$$
the numbers $A_2=A_2(n)$ and $B_2=B_2(n)$ which are the second
derivatives of the functions $E\Re\beta_j(r_n,\f)$ and
$E\Im\beta_j(r_n,\f)$ in the point $\f=\fb_n$, i.e.
$$
\aligned
& A_2=A_2(n)=-E\frac{r_n\xi\cos\fb_n}{U(r_n,\xi,\fb_n)}
-2E\frac{r_n^2\xi^2\sin^2\fb_n}{U(r_n,\xi,\fb_n)^2},\\
& B_2=B_2(n)=-E\frac{r_n\xi\sin\fb_n}{U(r_n,\xi,\fb_n)}
+2E\frac{r_n\xi\sin\fb_n(r_n\xi\cos\fb_n+r^2)}{U(r_n,\xi,\fb_n)^2},
\endaligned\tag 4.4
$$
$$
\aligned
\eta^{(2)}_j&=\eta^{(2)}_j(n)
=\left.\Re\frac {\partial^2}{\partial
\f^2}\(\beta_j(r_n,\f)-E\beta_j(r_n,\f)\)\right|_{\f=\fb_n}\\
&=-\frac{r_n\xi_j\cos\fb_n}{U(r_n,\xi_j,\fb_n)}
-2\frac{r_n^2\xi^2_j\sin^2\fb_n}{U(r_n,\xi_j,\fb_n)^2}-A_2,\\
\zeta^{(2)}_j&=\zeta^{(2)}_j(n)
=\left.\Im\frac {\partial^2}{\partial
\f^2}\(\beta_j(r_n,\f)-E\beta_j(r_n,\f)\)\right|_{\f=\fb_n}\\
&=\frac{-r_n\xi_j\sin\fb_n}{U(r_n,\xi_j,\fb_n)}
+2\frac{r_n\xi_j\sin\fb_n(r_n\xi_j\cos\fb_n+r_n^2)}
{U(r_n,\xi_j,\fb_n)^2}-B_2.
\endaligned  \tag4.5
$$
We can write
$$
\aligned
\Re \beta_j(r_n,\f)&=A_0+\eta_j^{(0)}(r_n,\fb_n)
+\eta_j^{(1)}(r_n,\fb_n)(\f-\fb_n) \\
&\qquad+\frac12\(A_2+\eta^{(2)}_j\)(\f-\fb_n)^2
+\frac16\vartheta_{j,1}(\f-\fb_n)^3,\\
\Im \beta_j(r_n,\f)&=B_0+\zeta_j^{(0)}(r_n,\fb_n)
+\zeta_j^{(1)}(r_n,\fb_n)(\f-\fb_n)\\
&\qquad+\frac12\(B_2+\zeta^{(2)}_j\)(\f-\fb_n)^2
+\frac16 \vartheta_{j,2}(\f-\fb_n)^3,
\endaligned
$$
where
$$
\aligned
\vartheta_{j,1}=\vartheta_{j,1}(r_n,\f)&=\left.\frac{\partial^3}
{\partial\f^3} \Re \beta_j(r_n,\f)\right|_{\f=\tilde\f},\\
\vartheta_{j,2}=
\vartheta_{j,2}(r_n,\f)&=\left.\frac{\partial^3}{\partial\f^3}
\Im \beta_j(r,\f) \right|_{\f=\tilde{\tilde\f}}
\endaligned \tag4.6
$$
with some numbers $\tilde\f$ and $\tilde{\tilde\f}$ in the interval
$[\f,\fb_n]$. Summing up the last relations for $j=1,\dots,n$, we get
the following relation for the function $Z_n(r_n,\f)$ defined in formula
(2.9):
$$
\aligned
Z_n(r,\f)&=n\(A_0+iB_0\)+\sqrt nS_0(n)+iT_0(n)
+\sqrt n(S_1(n)+iT_1(n))(\f-\fb_n)\\
&\qquad+\frac n2(A_2+iB_2)(\f-\fb_n)^2
+\frac{\sqrt n}2(\e_1(n)+i\e_2(n))(\f-\fb_n)^2\\
&\qquad+\frac{n}6(\delta_1(n)+i\delta_2(n))(\f-\fb_n)^3,
\endaligned \tag4.7
$$
where
$$
S_0=S_0(n)=\frac1{\sqrt n}\sum_{j=1}^n \eta^{(0)}_j(r_n,\fb_n)\quad
\text{and}\quad T_0=T_0(n)=\sum_{j=1}^n
\zeta^{(0)}_j(r_n,\fb_n)\;\text{mod }2\pi, \tag4.8
$$
$$
S_1=S_1(n)=\frac1{\sqrt n}\sum_{j=1}^n \eta^{(1)}_j(r_n,\fb_n),\qquad
T_1=T_1(n)=\frac1{\sqrt n}\sum_{j=1}^n \zeta^{(1)}_j(r_n,\fb_n),
\tag4.9 $$
and
$$
\align
\e_1(n)&=\frac1{\sqrt n}\sum_{j=1}^n \eta^{(2)}_j(r_n,\fb_n),\qquad
\e_2(n)=\frac1{\sqrt n}\sum_{j=1}^n \zeta^{(2)}_j(r_n,\fb_n)  \\
\delta_k(n)&=\frac 1n\sum_{j=1}^n \vartheta_{j,k}(r_n,\f),\quad
k=1,\;2.
\endalign
$$
 
We want to give a good asymptotic formula for the integral (2.8) by
means of formula (4.7) if Property~B holds. Define the intervals
$$
\bar I(n)=\[\fb_n-n^{-1/2+1/10},\fb_n+n^{-1/2+1/10}\]\quad\text{and}
\quad I(n)=\bar I(n)\cap [0,\pi).
$$
Observe that for sufficiently large $n$ $\bar I(n)=I(n)$ if
$0<\fb(\alpha^*)<\pi$, and $\bar I(n)=I(n)\cup(-I(n))$ if
$\fb(\alpha^*)=0$ or $\fb(\alpha^*)=\pi$ with
$\alpha^*=\lim\limits_{n\to\infty}\alpha(n)$. This relation follows from
Lemma~1, the relation $\lim\limits_{n\to\infty}\fb_n=\fb(\alpha^*)$
which we pointed out at the beginning of this Section, Property~B and
the observation that  in the case $\fb(\alpha^*)=0$ or $\pi$
$K^\pm(r(\alpha^*))>0$ with a strict inequality. Indeed, the inequality
$K^\pm(r(\alpha(n)))>0$ also holds in this case. These facts imply
the relation between the intervals $I(n)$ and $\bar I(n)$ formulated in
this paragraph.
 
We claim that there is an appropriate set $\Omega(n)$ on the probability
space where the random variables $\xi_1,\xi_2,\dots$ are defined such
that
$$
P(\Omega(n))\to 1\quad\text{as }n\to\infty, \tag4.10
$$
and
$$
\Re\(\frac1\pi \int_{I(n)}\exp\{Z_n(r_n,\f)\}\,d\f\)=\cases
2D_n&\text{if }0<\fb_n<\pi \\
D_n &\text{if }\fb_n=0 \text{ or }\fb=\pi
\endcases \tag4.11
$$
on the set $\Omega(n)$ for the function $Z_n(r,\f)$ defined in formula
(2.9) with a (random) number $D_n$ which satisfies the relation
$$
D_n=\Re\(\frac{\sqrt\pi \exp\left\{Z_n(r_n,\fb_n)-
\dfrac{(S_1(n)+iT_1(n))^2}
{2(A_2+iB_2)}
+O\(n^{-1/10}\)\right\}}{\sqrt{2n(-A_2-iB_2)}}\),
\tag$4.11'$
$$
where $S_1(n)$ and $T_1(n)$ are defined in
(4.9), $A_0$, $B_0$ in (4.3), $\fb_n=\fb(\alpha(n))$ and
$\sqrt{(-A_2-iB_2)}$ is meant as the square-root with positive real
part. Let us remark that $A_2<0$ which statement is proved with a
slightly different notation in Lemma~3. Moreover, the numbers $A_2(n)$
are strictly separated from zero for all sufficiently large $n$ since
$(r_n,\fb_n)\to(r(\alpha^*),\fb(\alpha^*))$ as $n\to\infty$, and
Property~A and Lemma~A can be applied if $\fb(\alpha^*)=0$
or $\fb(\alpha^*)=\pi$. We also claim that
$$
\align
&\text{the angle between the complex numbers }
\frac{\exp\left\{Z_n(r_n,\fb_n)-\dfrac{(S_1(n)+iT_1(n))^2}
{2(A_2+iB_2)}\right\}}{\sqrt{2n(-A_2-iB_2)}}\\
&\qquad\qquad\qquad \text{ and } i=\sqrt{-1} \text{ is
larger than } n^{-1/20}, \tag$4.11''$
\endalign
$$
and
$$
\left|\int_{[0,\pi]\setminus\(I(n)\)}\exp\{Z_n(r_n,\f)\}\,d\f\right|=
O\(\exp\left\{\Re Z_n(r_n,\fb_n)-\,\text{const.}\,n^{1/5}\right\}\),
\tag4.12
$$
and the $O(\cdot)$ is uniform in (4.11) and (4.12) on the sets
$\Omega(n)$.
 
Before the proof of relations (4.10), (4.11), $(4.11')$, $(4.11'')$ and
(4.12) we show that they imply Lemma~4.
First we show by a comparison of the right-hand side of (4.11),
$(4.11')$, $(4.11'')$ and (4.12) that a negligible error is committed on
the set $\Omega(n)$ if the integral (2.8) is restricted to the set
$I(n)$, i.e.\ the expression $D_n$ or $2D_n$ defined in $(4.11')$ is a
good approximation of $S^{(k)}(n)$.
 
Formula (4.15) which will appear in the definition of the set
$\Omega(n)$ implies that
$$
\left|\dfrac{(S_1(n)+iT_1(n))^2}
{2(A_2+iB_2)}\right|<\text{const.}\,n^{1/10}
$$
on the set $\Omega(n)$. This relation together with formulas $(4.11')$,
$(4.11'')$ imply that
$$
\align
|D_n|&\ge\text{const.}\left|\Re
\(\frac{\exp\left\{Z_n(r_n,\fb_n)-\dfrac{(S_1(n)+iT_1(n))^2}
{2(A_2+iB_2)}\right\}}{\sqrt{2n(-A_2-iB_2)}}\)\right|\\
&\ge n^{-1/20}\const
\left|\frac{\exp\left\{Z_n(r_n,\fb_n)-\dfrac{(S_1(n)+iT_1(n))^2}
{2(A_2+iB_2)}\right\}}{\sqrt{2n(-A_2-iB_2)}}\right|\\
&\ge \exp\left\{\Re Z_n(r_n,\fb_n)-\text{const.}\,n^{1/10}\right\}.
\endalign
$$
 
The above estimate together with (4.11), (4.12) and the definition of
$S^{(k)}(n)$ imply that $S^{(k)}(n)=2D_n(1+o(1))$ if
$0<\fb(\alpha^*)<\pi$ and $S^{(k)}(n)=D_n(1+o(1))$ if $\fb(\alpha^*)=0$
or $\fb(\alpha^*)=\pi$. Hence to prove Lemma~4 it is enough to give a
good estimate on $D_n$. We shall consider the cases
$0<\fb(\alpha^*)<\pi$, $\fb(\alpha^*)=0$ and $\fb(\alpha^*)=\pi$
separately. We get with the help of relation $(4.11')$ and the identity
$Z_n(r_n,\fb_n)=n\(A_0+iB_0\)+\sqrt nS_0(n)+iT_0(n)$  that on the set
$\Omega(n)$
$$
\align
D_n=\frac{\sqrt2}{\sqrt{Kn\pi}}\exp\left\{nA_0
+\sqrt nS_0-U_1\right\}&\cos\(nB_0+T_0-U_2-\frac\omega2\)
\(1+O(n^{-1/10})\)  \\
&\quad \text{if } 0<\fb(\alpha^*)<\pi \tag4.13
\endalign
$$
with
$$
U_1=U_1(n)=\frac{A_2(S_1^2-T_1^2)+2B_2S_1T_1}{2(A_2^2+B_2^2)},\quad
U_2=U_2(n)=\frac{-B_2(S_1^2-T_1^2)+2A_2S_1T_1}{2(A_2^2+B_2^2)},
\tag4.14
$$
and
$$
K=K(n)=(A_2^2+B_2^2)^{1/2}, \quad \omega=\omega(n)=\arctan
\frac{B_2}{A_2}, \tag$4.14'$
$$
because of the relation
$$
\frac{(S_1+iT_1)^2}{2(A_2+iB_2)}=U_1+iU_2. \tag$4.14''$
$$
In the case $\fb(\alpha^*)=0$, $B_0=0$, $B_2=0$, $T_0=0$ and $S_1=0$,
hence
$$
\aligned
D_n&=\frac{1}{\sqrt{2\pi|A_2|n}}\exp\left\{\frac{T_1^2}{2A_2}
+nA_0+\sqrt n S_0\right\}\(1+O(n^{-1/10})\)\quad\text{if
}\fb(\alpha^*)=0,
\endaligned \tag$4.13'$
$$
and in the case $\fb(\alpha^*)=\pi$,  $nB_0=n(-\pi-\alpha(n))=k(n)\pi$,
$T_0=0$ and $S_1=0$. Hence
$$
D_n=(-1)^{k(n)}\frac{1}{\sqrt{2\pi|A_2|n}}\exp\left\{\frac{T_1^2}
{2A_2} +nA_0+\sqrt n S_0\right\}\(1+O(n^{-1/10})\)
\quad\text{if }\fb(\alpha^*)=\pi.
\tag$4.13''$
$$
Lemma 4 follows from formulas (4.13), $(4.13')$, $(4.13'')$ and the
relation between $S^{(k)}(n)$ and $D_n$.
 
We define $\Omega(n)$ in the form $\Omega(n)=\Omega_1(n)
\cap\Omega_2(n)$. $\Omega_1(n)$ is the set where the above
relations hold:
$$
\aligned
&|S_1(n)|<n^{1/20},  \\
&|T_1(n)|<n^{1/20},  \\
&|\e_k(n)|<n^{1/10},\quad k=1,2,\\
&|\bar\delta_k(n)|<n^{1/10},\quad k=1,2,\\
&\biggl|\sum_{j=1}^n\frac{\xi_j\cos(\fb_n\pm n^{-4/10})}
{r_n^2+\xi_j^2+2r_n\xi_j\cos(\fb_n\pm n^{-4/10})}\\
&\hskip3truecm-nE\frac{\xi\cos(\fb_n\pm
n^{-4/10})}{r_n^2+\xi^2+2r_n\xi\cos(\fb_n\pm
n^{-4/10})}\biggr|<n^{11/20},
\endaligned \tag4.15
$$
where
$$
\bar\delta_k(n)=\frac1n \sum_{j=1}^n \bar\vartheta_{j,k},\quad
k=1,\;2
$$
with
$$
\align
\bar\vartheta_{j,1}&=\sup_{|\f-\fb_n|<n^{-1/2+1/10}}
\left|\frac12\frac{\partial^3}{\partial\f^3}
\(\log\(r_n^2+\xi^2_j+2r_n\xi_j\cos \f\)\)\right|\\
\bar\vartheta_{j,2}&=\sup_{|\f-\fb_n|<n^{-1/2+1/10}}
\left|\frac{\partial^3}{\partial\f^3}
\arccos\frac{r_n\cos\f+\xi}{(r_n^2+\xi^2+2r_n\xi\cos\f)^{1/2}}
\right|.
\endalign
$$
The set $\Omega_2(n)$ is defined as the set where the above relation
holds:
$$
\aligned
\left|W(n)-\frac12\right|>n^{-1/20}\quad\text{with } &W(n)=\frac1\pi\(
nB_0(n)+T_0(n)-U_2(n)-\frac{\omega(n)}2\)\;\text{mod }1 \\
&\qquad\text{if } 0<\fb(\alpha^*)<\pi,
\endaligned \tag$4.15'$
$$
where $B_0$, $T_0$, $U_2$ and $\omega$ are defined in (4.1), (4.2),
(4.14) and $(4.14')$.
 
The above defined set $\Omega(n)$ satisfies relation (4.10), since both
$\Omega_1(n)$ and $\Omega_2(n)$ satisfy it. It holds for $\Omega_1(n)$
since the random variables $\sqrt n S_1(n)$, $\sqrt n T_1(n)$,
$\sqrt n \e_k(n)$, $k=1,2$, and the last expression in (4.15) are
sums of $n$ independent random variables with expectation zero and
finite second moment, while $n\bar \delta_k(n)$ is the sum of $n$
independent random variables with finite expectation. Hence we can
deduce relation (4.15) from the Chebisheff and Markov inequalities if we
know that the appropriate variances and expected value have a uniform
bound for all sufficiently large $n$. But this holds because of
relation (2.5) and the fact that
$z(\alpha^*)=r(\alpha^*)e^{i\fb(\alpha^*)}$ and
$z_n=r_ne^{i\fb_n}$ are separated from the real line if
$0<\fb(\alpha^*)<\pi$, they are separated from $-\bold \Sigma$ if
$\fb(\alpha^*)=0$, and from $\bold \Sigma$ if $\fb(\alpha^*)=\pi$. The
last observation is needed to check that the singularity of the random
functions in the point $r_n$ or $-r_n$ makes no problem.
 
The probability
of the event that relation $(4.15')$ holds tends to 1, as $n\to\infty$.
This follows from Proposition~B which will be proved in Section~5.
Indeed, it follows from Proposition~B that the random variables $W(n)$
converge in distribution to the uniform distribution if $n\to\infty$,
and this implies $(4.15')$. The above mentioned limit theorem holds
because the vectors $(T_0(n),S_1(n),T_1(n))$ converge in distribution
to a random vector $(T_0,S_1,S_2)$ such that $T_0$ is uniformly
distributed mod~1, and the vector $(S_1,S_2)$ is independent of $T_0$.
The limit distribution for $W(n)$ follows from this fact and the
definition of $W(n)$.
 
Formula $(4.11'')$ follows from $(4.14'')$ and $(4.15')$.
To prove relation (4.11) and $(4.11')$ in the case $0<\fb(\alpha^*)<\pi$
observe that by (4.7) and the definition of the set $\Omega(n)$
$$
\aligned
Z_n(r_n,\f)-Z_n(r_n,\fb_n)&=n(A_2+iB_2)\frac{(\f-\fb_n)^2}2+\sqrt
n(S_1(n)+iT_1(n))(\f-\fb_n)\\
&\qquad\qquad\qquad +O\(n^{-1/10}\) \\
&=\frac{n(A_2+iB_2)}2\(\f-\fb_n+\frac{S_1+iT_1}{\sqrt n(A_2+iB_2)}\)^2
-\frac{(S_1+iT_1)^2}{2(A_2+iB_2)}\\
&\qquad\qquad\qquad +O\(n^{-1/10}\)
\endaligned \tag4.16
$$
if $\f\in I(n)$ and $\omega\in\Omega(n)$, hence
$$ \allowdisplaybreaks
\align
\int_{I(n)}\exp&\{Z_n(r_n,\f)-Z_n(r_n,\fb_n)\}\,d\f \\
&=\int_{I(n)}\exp\biggl\{
\frac{n(A_2+iB_2)}2\(\f-\fb_n+\frac{S_1+iT_1}{\sqrt n(A_2+iB_2)}\)^2 \\
&\qquad\qquad\qquad-\frac{(S_1+iT_1)^2}{2(A_2+iB_2)}+O\(n^{-1/10}\)
\biggr\}\,d\f \\
&=\int_{-\infty}^\infty\exp\biggl\{
\frac{n(A_2+iB_2)}2\(\f-\fb_n+\frac{S_1+iT_1}{\sqrt n(A_2+iB_2)}\)^2\\
&\qquad -\frac{(S_1+iT_1)^2}{2(A_2+iB_2)}+O\(n^{-1/10}\)\biggr\}\,d\f
+O\(e^{-K n^{1/5}}\)\\
&=\frac{\sqrt {2\pi}\exp\left\{-\dfrac{(S_1+iT_1)^2}{2(A_2+iB_2)}
+O\(n^{-1/10}\)\right\}} {\sqrt{(-A_2-iB_2)n}},
\endalign
$$
since $\int_{\infty}^\infty e^{-A(\f-B)^2}\,d\f=\sqrt{\frac \pi A}$ if
$\Re A>0$, and the main term in the middle expression of the last
relation is dominating being larger than $O\(e^{-\const n^{1/10}}\)$.
In the above calculation we have exploited that $A_2<0$. The
expression $\sqrt{(-A_2-iB_2)}$ is meant as the square-root with
positive real part.
 
The cases $\fb(\alpha^*)=0$ or
$\fb(\alpha^*)=\pi$ are similar, but simpler. The integrals we are
interested in can be calculated similarly, only the approximating
integrals $\int_{-\infty}^\infty$ must be replaced by $\int_0^\infty$ or
$\int_{-\infty}^0$. (We exploit during these calculations that $S_1=0$
in the present case.) The main part of the integral under
consideration is real, since $S_1=0$, $B_2=0$, $T_0=0$ and
$B_0=0$~mod~$\pi$ in this case.
 
To prove (4.12) it is enough to show that
$$
\Re Z_n(r_n,\f)\le \Re Z_n(r_n,\fb_n)-\,\text{const.}\,n^{1/5}\quad
\text{if }\f\in [0,\pi]\setminus I(n)  \tag4.17
$$
on the set $\Omega(n)$, where the function $Z_n(r_n,\f)$ is defined in
formulas (2.9) and (2.10). First we show the following weaker result:
$$
\Re Z_n\(r_n,\fb_n\pm n^{-2/5}\)<\Re
Z_n\(r_n,\fb_n\)-\text{const.}\,n^{1/5}, \tag4.18
$$
i.e.\ relation (4.17) holds if some very special points of the set
$[0,\pi]\setminus I(n)$ are considered.
 
To prove relation (4.18) let us first observe that for $A_2=A_2(n)$
defined in (4.4)  $A_2<-K$ with some negative constant $K$. Indeed,
either $0<\fb<\pi$ in which case
$A_2=-2E\dfrac{r_n^2\xi^2\sin^2\fb_n}{U(r_n,\xi,\fb_n)^2}<-K$ because
of Lemma~1 or $\fb_n=0$ or $\fb_n=\pi$, and in these cases
$A_2=E\dfrac{\mp r_n\xi}{(r_n\pm \xi)^2}<-K$ because of Property~B. We
get relation (4.18) by taking the real part of the first identity in
(4.16) with the choice $\f=\fb_n\pm n^{-2/5}$ with the help of the
following observations:
$nA_2\dfrac{(\f-\fb_n)^2}2<-\text{const.}\,n^{1/5}$, $\sqrt
n|(\f-\fb_n)S_1(n)|<n^{3/20}$ on the set $\Omega(n)$ because of the
relation $A_2<-K$ and formula~(4.15).
 
Relation (4.17) can be rewritten, with the change of
variable $\psi=\cos\f$, in the equivalent form
$$
Y_n(\psi)\le Y_n(\cos\fb_n)-\,\text{const.}\,n^{1/5}\quad\text{if
}|\arccos \psi-\fb_n|\ge n^{-2/5},  \tag4.19
$$
on the set $\Omega(n)$, with the function $Y_n(\psi)$ defined as
$$
Y_n(\psi)=\Re Z_n(r_n,\arccos\psi)=\sum_{j=1}^n
\frac12\log\(r_n^2+\xi^2_j+2r_n\xi_j\psi\) -n\alpha(n)\log r_n.
$$
Relation (4.18) implies that
$$
Y_n\(\cos(\fb_n\pm n^{-2/5})\)\le
Y_n(\cos\fb_n)-\,\text{const.}\,n^{1/5}. \tag4.20
$$
To prove (4.19) it is enough to observe that
$$
\frac{d^2}{d \psi^2} Y_n(\psi)=-\sum_{j=1}^n
\frac{2r_n^2\xi_j^2}{\(r_n^2+\xi^2_j+2r_n\xi_j\psi\)^2}\le 0, \tag4.21
$$
hence the function $Y_n(\cdot)$ is concave, and relation (4.20) implies
its strengthened form, relation (4.19).
 
\beginsection 5. Proof of the limit theorems for sums of independent
vectors.
 
{\it Proof of Proposition A.}\/ In the proof we apply a natural
adaptation of the characteristic function technique. We shall
investigate the expressions
$$
\f(t,l)=E\exp\{itX_1+2\pi il(Y_1-\alpha)\}, \tag5.1
$$
where $t\in R^k$, $l$~is an arbitrary integer if $G_0=G$, $l$ is an
integer, $0\le l<p$ if $G_0=\left\{\dfrac jp,\;j=0,\dots,p-1\right\}$,
and $tX_s$ denotes scalar product. We claim that
$$
\aligned
\f(t,l)&=\exp\left\{-\frac12t\Sigma
t^*+o(t^2)\right\}\quad\text{if }l=0\text{ and }t\to0 \\
|\f(t,l)|&<C<1  \quad\text{if }l\neq 0 \text
{ and } |t|<\e, \endaligned \tag5.2
$$
where the constants $C<1$ and $\e>0$ may depend on $l$,
 
Since $EX_1=0$, and the coefficient of $Y_1-\alpha$ in the definition of
the function $\f(t,l)$ is zero for $l=0$, the first line of relation
(5.2) follows from a simple Taylor expansion, just as it is done in the
proof of the classical central limit theorem. First we prove the second
line of (5.2) first in the case if $G_0=G$, i.e.\ if the minimal coset
containing the support $\mu$ is the whole group $G$.
We show that in this case for all positive integers $l$ and $0\le
\alpha\le 1$ there is some $\delta=\delta(l)>0$ and $\eta=\eta(l)>0$
depending only on $l$ such that the distribution $\mu$ of $Y_s$
satisfies the inequality
$$
\mu\(\bigcup_{j=1}^l \[\frac jl-\eta+\alpha,\frac
jl+\eta+\alpha\]\)<1-\delta. \tag5.3
$$
Let us emphasize that the numbers $\eta>0$ and $\delta>0$ in formula
(5.3) may depend on $l$ but not on $\alpha$.
 
To prove (5.3) first we show that for all sets
$$
A(\beta)=A(\beta,l,\eta)=\bigcup\limits_{j=1}^l\(\dfrac
jl+\beta-2\eta,\dfrac jl+\beta+2\eta\),
$$
$\mu(A(\beta))<1-\delta$ if the numbers $\eta=\eta(\beta,l)$ and
$\delta=\delta(\beta,l)$ are appropriately chosen. Indeed, the $\mu$
measure of the (finite) sets $\bigcup\limits_{j=1}^l\left\{\dfrac
jl+\beta\right\}$ is less than one for all $0\le\beta\le1$, since
otherwise the support of the measure $\mu$ were concentrated on a
finite coset. Since these sets are compact, this relation also holds
for their sufficiently small neighbourhoods.
 
Since the group $G$ is compact, there is a finite cover of $G$ with
some sets of the form $\bar A(\beta)$, which sets are defined in the
same way as $A(\beta)$, ($\mu(A(\beta)<1-\delta(\beta)$), only
$2\eta$ is replaced by $\eta$ in their definition. If we choose $\eta$
as the minimum of the numbers $\eta$ appearing in the definition of the
sets $\bar A(\beta)$ appearing in this finite cover, then  all sets
which are considered at the left-hand side of (5.3) are contained in
one of the sets $A(\beta)$ for which $\bar A(\beta)$
takes part in this cover. Hence relation (5.3) holds if $\eta$ and
$\delta$ are chosen as the minimum of those values $\eta(\beta)>0$ and
$\delta(\beta)>0$ which appear in the sets $A(\beta)$ for which
$\bar A(\beta)$ takes  part in this finite cover of $G$.
 
Relation (5.3) implies that for sufficiently small $\e=\e(l)>0$
$$
P\(|l Y_1+tX_1-\alpha|>\frac \eta2\)>\frac \delta2
$$
for all $0\le\alpha\le1$ if $t<\e$ with some $\eta=\eta(l)$ and
$\delta=\delta(l)$. Hence
$$
E\Re e^{i(lY_1+tX_1-\alpha)}<1-\frac {\eta\delta^2}8
$$
for all $\alpha\in[0,1]$. Since this relation holds for all $\alpha$,
this implies the second line of (5.2) in the case $G=G_0$.
 
If $G_0+\alpha$ with $G_0=\left\{\dfrac jp,\;j=0,1,\dots,p-1\right\}$,
is the minimal coset containing the support of $\mu$, then the
distribution of $Y_1-\alpha$ is concentrated on $G_0$. The distribution
of $Y_1-\alpha$ is concentrated on some points of the form
$\dfrac{k_u}p$, $u=0,\dots, r$ with some $r$ such that the $\mu$ measure
of all these points $\dfrac{k_u}p$ is positive. We may assume by
replacing $\alpha$ by $\alpha-\dfrac{k_0}p$, if this is needed, that
$k_0=0$. Moreover, because of the minimality property of $G_0$ the
greatest common divisor of $k_1,\dots,k_r$ and $p$ equals 1. Hence
there are some integers $N_u$, $u=1,\dots,r$  and $N$ such that
$$
Np+\sum_{u=1}^r N_uk_u=1. \tag5.4
$$
This fact implies that for any $1\le l<p$ all vectors $\exp\left\{2\pi
il\dfrac {k_u}p\right\}$, $u=0,\dots,r$ cannot be parallel. Indeed,
otherwise the relation $l k_u=lk_0=0$ mod~$p$ would hold for all
$u=1,\dots,r$, and this contradicts to~(5.4). Also the maximum between
the angles of the vectors $\exp\left\{itX_1+2\pi \dfrac
{ilk_u}p\right\}$ are separated from zero with positive probability, and
this fact implies the second line of (5.2) in this case, too.
 
Since $E\exp\{itU_n+2\pi ilV_n\}=\[ \f\(\dfrac
t{\sqrt n}X_1,l\)\]^n$, relation (5.2) implies
that
$$
E\exp\{itU_n+2\pi ilV_n\}=
\cases
\exp\left\{-\dfrac12t\Sigma t^*\right\}(1+o(1))&\text{if }l=0\\
o(1)&\text{if } l\neq0,
\endcases
$$
Here $t\in R^k$,  $l=0,\pm1,\pm2,\dots$ if $G_0=G$, and $l$ is an
integer, $0\le l<p$, if $G_0=\left\{\dfrac jp,\;j=0,\dots,p-1\right\}$.
This means that $\lim\limits_{n\to\infty}E\exp\{itU_n+2\pi
ilV_n\}=E\exp\{itU+2\pi ilV\}$ for all such $t$ and
$l$, where $(U,V)$ is such a random vector whose distribution is
described in Proposition~A. This relation implies the Proposition~A.
\medskip \noindent
{\it Proof of Proposition B.}\/ The proof is a slight modification of
that of Proposition~A. It is enough to prove a modification of (5.2)
under the condition of Proposition~B where the characteristic function
$\f(t,l)$ is replaced by $\f_n(t,l)=E\exp\{itX_1(n)+2\pi
il(Y_1(n)-\alpha)\}$. The constant $C<1$ in the second line of this
modified relation (5.2) must not depend on~$n$. The first line of this
modified formula (5.2) holds, since it holds if $X_1(n)$ is replaced by
$X$, and $\(Ee^{itX_1(n)}-Ee^{itX}\)=o(t^2)$ as $t\to0$. The second
line of (5.2) can be deduced from a modified version of formula (5.3)
where the distribution $\mu$ of $Y_1$ is replaced by the distribution
$\mu_n$ of $Y_1(n)$, but the numbers $\eta$ and $\delta$ must not
depend on $n$. This can be deduced, just as it was done in the proof
of Proposition~A, from the weaker relation $\mu_n(A(\beta))<1-\delta$
with
$$
A(\beta)=A(\beta,l,\eta)=\bigcup\limits_{j=1}^l\(\dfrac
jl+\beta-2\eta,\dfrac jl+\beta+2\eta\),
$$
$\delta>0$, $\eta>0$, if the numbers $\eta=\eta(\beta,l)$ and
$\delta=\delta(\beta,l)$ are appropriately chosen. We have already
proved in Proposition~A that $\mu(A(\beta))<1-\delta$, where $\mu$ is
the (weak) limit of the measures $\mu_n$. Moreover, this statements also
holds for the closure $\bar A(\beta)$ of the set $A(\beta)$ with a
possibly smaller parameter $\eta$. Since $\mu_n\Rightarrow \mu$,
$\limsup\limits_{n\to\infty}\mu_n(\bar A(\beta))\le \mu(\bar A(\beta))$.
This implies that also the relation $\mu_n(A(\beta))<1-\delta$ holds for
large $n$. Proposition~B is proved.
 
\beginsection 6. The proof of the main results.
 
{\it Proof of Theorem 1.}\/ By Lemma~4 $\log
\left|S^{(k)}(n)\right|-\log\left|\bar S^{(k)}(n)\right|\Rightarrow 0$,
where $\bar S^{(k)}(n)$ is defined in (2.12), and $\Rightarrow$ denotes
stochastic convergence. Hence $S^{(k)}(n)$ can be replaced by $\bar
S^{(k)}(n)$ in the proof of Theorem~1.
 
We claim that
$$
\aligned
&\frac{U_1(n)}{\sqrt n}\Rightarrow 0,\quad
\frac{T_1^2(n)}{\sqrt n}\Rightarrow 0\quad
\text{and}\\
&\frac1{\sqrt n}
\log\left|\cos\(nB_0(n)+T_0(n)-U_2(n)-\frac{\omega(n)}2\)\right|
\Rightarrow0.
\endaligned \tag6.1
$$
The third relation in (6.1) is needed only in the case when
$0<\f(\alpha^*)<\pi$. The first two relations in (6.1) are trivial,
since the random variables $U_1(n)$ and $T_1^2(n)$ are stochastically
bounded. They are even stochastically convergent. The third relation
holds, since the random variables $T_0(n)-U_2(n)$~mod~$2\pi$ converge
in distribution to the uniform distribution in $[0,2\pi)$. Indeed, by
Proposition~B the random vectors $(T_0(n),U_2(n))$ converge in
distribution to a random vector $(T,U)$, where $T$ and $U$ are
independent and $T$ is uniformly distributed in $[0,2\pi)$. Hence the
random variables $T_0(n)-U_2(n)$~mod~$2\pi$ converge in distribution to
the uniform distribution of $[0,2\pi)$, as we claimed. This relation
implies that the random variables
%$\log\left|\cos\(n(B_1-\alpha(n)\f)+T_1(n)-U_2(n)-\dfrac{\omega(n)}2\)\right|$
$\log\left|\cos\(n(B_0(n)-U_2(n)-\dfrac{\omega(n)}2\)\right|$
converge in distribution to a random variable $\log|\cos V|$, where $V$
is uniformly distributed in~$[0,2\pi)$. This implies that the third
relation also holds in (6.1). The random variables $S_0(n)$ converge to
a normal law with expectation zero and variance Var$\,\eta$, and a
slight refinement of the previous argument
also shows that the vectors
$$
\(\log\cos\(n(B_0(n)-U_2(n)-\dfrac{\omega(n)}2\),S_0(n)\)
$$
converge in distribution to a
random vector $(\log \cos V, Z)$, where $V$ and $Z$ are independent
random variables, $V$ is uniformly distributed in $[0,2\pi]$, and $Z$ is
normally distributed with expectation zero and variance
$\text{Var}\,\eta$. Relation (2.13) follows from the above
observations. Because of Lemma~4, the form  of $\bar S^{(k)}(n)$
defined in (2.12) and the limit behaviour of the expression
in the  second relation of (6.1) the sign of $S^{(k)}(n)$ also satisfies
the relations given in Theorem~1. \medskip\noindent
{\it Proof of Lemma 5.}\/ The random variable $\eta=\eta(\alpha^*)$ is
constant if and only if
$$
\xi^2+2r(\alpha^*)\xi\cos\fb(\alpha^*)=\,\text{const.}\quad\text{
with probability 1.}
$$
Since $\xi$ is a non-constant random variable, and its values satisfy
an equation of second order, its distribution is concentrated in two
points $x_1$ and $x_2$ which satisfy the identity
$x_1^2+2r(\alpha^*)x_1\cos\fb(\alpha^*)=x_2^2+2r(\alpha^*)x_2
\cos\fb(\alpha^*)$, or equivalently
$x_1+x_2+2r(\alpha^*)\cos\fb(\alpha^*)=0$. In case a.)
when the relation $0<\fb(\alpha^*)<\pi$ holds, by Lemma~1 the identity
$E\dfrac\xi{r^2(\alpha^*)+\xi^2+2r(\alpha^*)\xi\cos\fb(\alpha^*)}=0$
must hold. This is equivalent to the relation $px_1+qx_2=0$ with
$p=P(\xi=x_1)$, $q=P(\xi=x_2)=1-p$, since
$r^2+x_1^2+2rx_1\cos\fb=r^2+x_2^2+2rx_2\cos\fb$
in this case. Finally, the second equation of the fixed point equation
(1.4) \ $r\left.\dfrac{\partial H}{\partial
r}\right|_{r=r(\alpha^*)}=\alpha^*$
yields that $E\dfrac{r\xi\cos\fb+r^2}{r^2+\xi^2+2r\xi\cos\fb}=\alpha^*$.
This is equivalent to $\dfrac {r^2}{r^2-x_1x_2}=\alpha^*$, since in
this case $r^2+\xi^2+2r\xi\cos\fb=r^2-x_1x_2$, as the calculation
$r^2+\xi^2+2r\xi\cos\fb=r^2+(px_1^2+qx_2^2)
=r^2+(px_1+qx_2)(x_1+x_2)-x_1x_2=r^2-x_1x_2$ shows.
 
We have proved that the distribution of the random variable $\xi$ must
be concentrated in two different points, and the above equations make
possible to calculate $r(\alpha^*)$ and $\fb(\alpha^*)$ from $\alpha^*$.  To decide
whether we get a real solution for a pair $(F,\alpha^*)$ we have to
check whether the condition $|\cos(\f(\alpha^*)|<1$ is satisfied. Some
calculation shows that $\cos\fb(\alpha^*)=-\dfrac{x_1+x_2}{2r(\alpha^*)}
=-\dfrac{(q-p)x_1}{2qr(\alpha^*)}$, $r(\alpha^*)^2=\dfrac
pq\dfrac{\alpha^*}{1-\alpha^*}x_1^2$. The last two identities yield that
$\cos^2\fb(\alpha^*)=\dfrac{(p-q)^2}{4pq}\dfrac{1-\alpha^*}{\alpha^*}$.
This gives that the condition $|\cos\fb(\alpha^*)|<1$ is equivalent to
$\alpha^*>1-4pq$.
 
In case b.) when the relation $\fb(\alpha^*)=0$ holds the random
variable $\xi$ is concentrated in two points $x_1$, $x_2$, \
$x_1+x_2+2r(\alpha^*)=0$, and
$E\dfrac\xi{(r+\xi)^2}\ge0$. The latter relation is equivalent to
$E\xi\ge 0$ in the present case. Since
$2r(\alpha^*)=-(x_1+x_2)$ the second part of the fixed point equation
(1.4) yields that $\alpha^*=E\dfrac
r{r+\xi}=-\dfrac{(p-q)(x_1+x_2)}{x_1-x_2}$.
The conditions $px_1+qx_2\ge0$, $x_1+x_2<0$ are satisfied. The last
condition appears, because it is equivalent to $r(\alpha^*)>0$. Some
calculation shows that under such conditions the relation $0<\alpha^*<1$
also holds. Case c.) in Lemma~5 when $\fb(\alpha^*)=\pi$ can be handled
similarly to case~b.). Lemma~5 is proved. \medskip\noindent
{\it Proof of Theorem 2.}\/ Because of Lemma~4 the random variable
$S^{(k)}(n)$  can be replaced by $\bar S^{(k)}(n)$ defined in the first
line of formula (2.12) in the proof of the limit theorem. Moreover,
under the conditions of Theorem~2 $\sqrt{n} S_0(n)=0$, i.e.\ this term
is missing from formula (2.12). Proposition~B implies that the random
vectors
$$
\(-U_1(n),\;nB_0+T_1-U_2-\dfrac\omega2\;\;\text{mod }2\pi\). \tag6.2
$$
converge in distribution to a random vector $(U,Z)$, where
$Z=Z_1-U_2+\text{const.}\,\mod 2\pi$ with $U_2=\dfrac
{-B_2(S^2-T^2)+2A_2ST}{2(A_2+B_2^2)}$, $U_1=-U=-\dfrac
{A_2(S^2-T^2)+2B_2ST}{2(A_2+B_2^2)}$, $(S,T)$ is a Gaussian random
vector with expectation zero and covariance matrix given in (2.14), the
random variable $Z_1$ is uniformly distributed in $[0,2\pi)$, and it is
independent of the vector $(S,T)$. These relations imply that the random
variable $Z$ is also uniformly distributed in $[0,2\pi)$, and it is
independent of the vector $(S,T)$ hence also of the random variable
$U$, since its conditional distribution under the condition $S=x$,
$T=y$ is the uniform distribution on $[0,2\pi)$ for all $x$ and $y$.
Lemma~4 together with the convergence of the random vectors defined in
(6.2) in distribution to the random vector $(U,Z)$ imply
Theorem~2.\medskip\noindent
{\it Proof of Theorem $2'$.}\/ Here again the investigation of the
random variable $S^{(k)}(n)$ can be replaced by that of $\bar
S^{(k)}(n)$ defined in the second line of formula (2.12).
We are interested in the asymptotic behaviour of the expression in the
exponent of this formula. We describe the central limit theorem for the
random vector $(L_n^{-1}S_0(n),T_1(n))$ with the definition of an
appropriate normalization $L_n$.
 
We have $\sqrt n S_0(n)=\sum\limits_{j=1}^n(\eta^{(0)}_j-E\eta^{(0)}_j)$
with $\eta^{(0)}_j=\log |r(\alpha(n))+\xi_j|$. Under the conditions of
Theorem~$2'$ $\lim\limits_{n\to\infty}\text{Var}\,\eta_j(n)=0$, but to
determine the right norming $L_n$ we need a sharper estimate on this
variance. To get it, observe that $r(\alpha(n))=
r(\alpha^*)+(\alpha(n)-\alpha^*)r'(\alpha^*)+O\((\alpha(n)-\alpha^*)^2\)$,
and since $x_1+x_2+2r(\alpha^*)=0$,
$\eta_j\sim\log\left|\xi_j-\dfrac{x_1+x_2}2
+r'(\alpha^*)(\alpha(n)-\alpha^*)\right|$. Hence $\eta_j$ takes two
values $y_1$ and $y_2$ with probabilities $p$ and $q$, and
$|y_1-y_2|=\dfrac{4r'(\alpha^*)|\alpha(n)-\alpha^*|}{x_1-x_2}(1+o(1))$,
where $x_1>x_2$. We get with the help of some calculation from the
second relation in (1.4) and the relations $\fb(\alpha)=0$ in a small
neighbourhood of $\alpha^*$ that
$r'(\alpha^*)E\dfrac{\xi}{(r+\xi)^2}=1$. Because of this identity and
the relation $x_1+x_2+2r(\alpha^*)=0$
that $r'(\alpha^*)=\dfrac {(x_1-x_2)^2}{4(px_1+qx_2)}$. Hence
Var$S_0(n)=\text{Var}\,\eta_j=pq(y_1-y_2)^2\sim
pq(\alpha(n)-\alpha^*)^2\dfrac{(x_1-x_2)^2}{(px_1+qx_2)^2}$. On the
other hand, some calculation yields that
Var$\,T_1(n)=\dfrac{(x_1+x_2)^2}{(x_1-x_2)^2}$. Since the random
variables $\xi_j$ take two values, the random variables $S_0(n)$ and
$T_1(n)$ are linear transform of each other. Because of the above
observations and the central limit theorem the random vectors
$(L_n^{-1}S_0(n),T_1(n)$ converge in distribution to a vector
$\(V,\dfrac{x_1+x_2}{x_1-x_2}V\)$ with the choice
$L_n=\sqrt{pq}|\alpha(n)-\alpha^*|\dfrac{x_1-x_2}
{px_1+qx_2}$,  where $V$ is a standard normal random variable. This
limit theorem together with the form of the second line in formula
(2.12) imply Theorem~$2'$.
\medskip\noindent
{\it Acknowledgement:}\/ The author would thank the referee for some
useful remarks which helped to simplify certain technical details. In
particular, the formulation and proof of Lemma~A and some simplification
in the proof of Lemmas~2 and~3 were based on his ideas.
 
\parskip=1.5pt
\smallskip \noindent {\bf References:}
\smallskip
\item{1.)} Dynkin, E. B., Mandelbaum, A. (1983) Symmetric statistics,
Poisson processes and multiple Wiener integrals.  {\it Annals of
Statistics\/} {\bf 11} 739--745
\item{2.)} van Ess, G. (1986) On the weak limit of elementary symmetric
polynomials  {\it Annals of Probability}\/ {\bf 14} No.\ 2 667--695
\item{3.)} Hal\'asz, G. and Sz\'ekely, G. J. (1976) On the
elementary
symmetric polynomials of independent random variables {\it Acta Math.
Acad. Sci. Hungar.}\/ {\bf 28} 397--400
\item{4.)} M\'ori, T. F. and Sz\'ekely, G. J. (1982) Asymptotical
behaviour
of symmetric polynomial statistics {\it The Annals of Probability}\/
{\bf 10} No.\ 1 124--131
\item{5.)} Parthasarathy, K. R. Probability Measures on Metric Spaces.
{\it Academic Press,}\/ New York and London, 1967
\item{6.)} Raugi, A. (1979) Th\'eor\`eme de la limite centrale pour
un produit semi-direct d'un groupe de Lie r\'esoluble simplement
connexe de type rigide par un groupe compact. In: H. Heyer (ed.)
Probability Measures on Groups. Proceedings, Oberwolfach, 1978.
(Lecture Notes in Math., vol.~706, pp.~257--324) Springer, Berlin
Heidelberg New York.
 
 
\bye
 
 
