THE LIMIT BEHAVIOUR OF ELEMENTARY SYMMETRIC POLYNOMIALS OF I.I.D. RANDOM VARIABLES WHEN THEIR ORDER TENDS TO INFINITY

Péter Major Mathematical Institute of the Hungarian Academy of Sciences and Bolyai College of the Eötvös Loránd University, Budapest

Abstract: Let ξ_{1}, ξ_{2}, … 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 ξ_{1}, …, ξ_{n} of this sequence. We are interested in the limit behaviour of S^{(k)}(n) with an appropriate transformation if \frac{k(n)}{n} → α, 0 < α < 1. Since k(n) → ∞ as n → ∞, 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.

1. Introduction

In this paper the following problem is investigated: Let ξ_{1}, …, ξ_{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 ξ_{j}, j = 1, …, n is not concentrated in a single point. Define the elementary symmetric polynomials

S^{(k)}(n) = S^{(k)}(n, ξ_{1}, …, ξ_{n}) = ∑_{1≤i_{1}<i_{2}<···<i_{k}≤n} ξ_{i_{1}} ⋯ ξ_{i_{k}}.  (1.1)

We are interested in the limit behaviour of the random variables S^{(k)}(n) if n → ∞,

k = k(n), and α(n) → α^{∗}, P (ξ = 0) < α^{∗} < 1, where α(n) = 1 − \frac{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 Eξ = 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 ξ_{j} are non-negative, and in paper [4] the limit behaviour of S^{(k)}(n) was described in the special case when P (ξ_{j} = 1) = P (ξ_{j} = −1) = 1/2.

The author got support from the Hungarian OTKA Foundation No. T26176.

Paper [2] contains a generalization of paper [4] when the distribution of ξ_{j} is concentrated in three point, 0 and ±1, and P (ξ_{j} = 1) = P (ξ_{j} = −1) = 1/2P (ξ_{j} ≠ 0). 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 α^{∗} = lim_{n→∞} α(n) or it strongly depends on the sequence k(n) and the distribution function F (x) of the random variables ξ_{j}. We prove that in the generic case, although the normalization depends on α(n), the limit distribution depends only on α^{∗}.

The investigation is based on the following observation. Define the polynomial

Z_{n}(x) = Z_{n}(x, ξ_{1}, …, ξ_{n}) = ∏_{j=1}^{n}(x + ξ_{j}).

Then

Z_{n}(x) = ∑_{k=1}^{n} S^{(k)}(n)x^{n−k},

hence

\begin{aligned}S^{(k)}(n) & = \frac{1}{(n − k)!} \frac{d^{(n−k)}}{dx^{(n−k)}} Z_{n}(x) = \frac{1}{2πi} ∮_{|ζ|=r} \frac{Z_{n}(ζ)}{ζ^{n−k+1}} dζ \\ & = \frac{1}{2π} ∫_{−π}^{π} \frac{∏_{j=1}^{n} |re^{iφ} + ξ_{j}|}{r^{n−k}} exp \{−i(n − k)φ + i ∑_{j=1}^{n} arg (re^{iφ} + ξ_{j})\} dφ \end{aligned} (1.2)

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, φ̄) = (r(ξ_{1}, …, ξ_{n}), φ̄(ξ_{1}, …, ξ_{n})) where the partial derivatives of the (random) expression

∑_{j=1}^{n} log |re^{iφ} + ξ_{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

∑_{j=1}^{n} log |re^{iφ} + ξ_{j}| ∼ nE log |re^{iφ} + ξ| = nH(r, φ)

with

H(r, φ) = H(z) = E log |re^{iφ} + ξ| = \frac{1}{2}E log (r^{2} + ξ^{2} + 2rξ cos φ) ,  (1.3)

where ξ is an F distributed random variable, and z = re^{iφ}. 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 ≤ φ ≤ π, i.e. for a solution in the upper half-plane. We will show that under general conditions there is a point (r, φ̄), φ̄ = φ̄(r), such that the relations

\frac{∂}{∂φ} [H(r, φ̄) − α(n) log r] = 0, \frac{∂}{∂r} [H(r, φ̄) − α(n) log r] = 0

hold. We rewrite these equations in the equivalent form

\frac{∂}{∂φ}H(r, φ) = 0|_{φ=φ̄} , r \frac{∂}{∂r} H(r, φ)|_{φ=φ̄} = α(n),  (1.4)

and also require that the solution (r, φ̄) satisfy the relation

φ̄ is the place of maximum of H(r, φ) (as a function of φ, 0 ≤ φ ≤ π.).  (1.5)

Let us remark that the solution of the equation (1.4) (together with the property (1.5) depends on n through the function α(n). Although this dependence on n will turn out to be weak in the case when lim_{n→∞} α(n) = α^{∗}, 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, φ̄) 0 ≤ φ̄ ≤ π 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.

2. The strategy of the proof

Consider the function H(r, φ) = 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 ξ_{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, φ). 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^{iφ} for which the number z has a non-zero imaginary part, i.e. for which φ ≠ 0 and φ ≠ π. On the other hand, for φ = 0 or φ = π 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 α(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^{±}(r) = E \frac{±ξ}{(ξ ± r)^{2}} , r > 0  (2.1)

and sets

A^{±} = {r: r > 0 and K^{±}(r) ≥ 0},  (2.2)

where ξ is an F (x) distributed random variable. Let us remark that the integral (2.1)

is always meaningful, although the relation E \frac{±ξ}{(ξ ± r)^{2}} = −∞ 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 ∈ A^{+} ∪ A^{−}.

We formulate the following property:

Property A. If r ∈ A^{+}, then there is a number h = h(r) > 0 such that the interval (−r − h, −r + h) has zero F measure. If r ∈ 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 Σ denote the support of the distribution of ξ, i.e. the smallest closed set on the real line R^{1} such that P (ξ ∈ Σ) = 1. (Such a set exists. See e.g. [5], Chapter 2, Theorem 2.1.) Then for all r ∈ A^{+} d(r, −Σ) > 0 and for all r ∈ A^{−} d(r, Σ) > 0.

Property A is less restrictive than it may seem in the first moment, because the sets A^{±} are small. Thus for instance, r ∉ A^{±} if the distribution function F has a non-zero density function in a neighbourhood of the point ∓r, or more generally if F (∓r + h) − F (∓r) > Cη^{2} or F (∓r) − F (∓r − h) > Cη^{2} with some C > 0, h > 0 and 0 < η < h. Indeed, K^{±}(r) = −∞ in this case. Thus Property A holds if for all x F (x + h) − F (x − h) ≥ 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 ξ random variable is symmetrically distributed, since the sets A^{±} are empty in this case. Indeed, in this case

K^{±}(r) = E \frac{±ξ}{(r ± ξ)^{2}} = \frac{1}{2}E (\frac{±ξ}{(r ± ξ)^{2}} + \frac{∓ξ}{(r ∓ ξ)^{2}} ) = −E \frac{2rξ^{2}}{((r^{2} − ξ^{2})^{2}} < 0

for all r > 0.

We also assume that

E|ξ| < ∞, and E \frac{1}{|ξ|}I(ξ ≠ 0) < ∞  (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 (ξ = 0) < α(n) < 1, then the equation (1.4) has a unique solution which satisfies (1.5).

Lemma 1. Fix some r > 0 and consider the function H(r, φ), defined in formula (1.3) as a function of φ, 0 ≤ φ ≤ π. (The function H(r, φ) can also take the value −∞ in the end points 0 and π.) The function H has a unique maximum at a value φ̄ = φ̄(r) defined by the formula

φ̄(r) = \{ \begin{matrix}0 & if E \frac{ξ}{(r + ξ)^{2}} ≥ 0 \\ π & if E \frac{ξ}{(r − ξ)^{2}} ≤ 0 \\ the unique solution of the equation E \frac{ξ}{r^{2} + ξ^{2} + 2rξ cos φ} = 0 (in the variable φ, 0 ≤ φ ≤ π) & if E \frac{ξ}{(r + ξ)^{2}} < 0 < E \frac{ξ}{(r − ξ)^{2}} . \end{matrix} (2.4)

The relation

\frac{∂H(r, φ)}{∂φ}|_{φ=φ̄} = 0  (2.5)

holds.

Define the function E(r, φ) = r \frac{∂}{∂r} H(r, φ) and G(r) = E(r, φ̄(r)).

Lemma 2. G(r) is a continuous and strictly monotone increasing function.

Before the proof of Lemma 2 we prove the following technical Lemma A.

Lemma A. The function H(z) defined in formula (1.3) is analytic in the set C \ (−Σ) and the functions K^{±}(z), the analytical continuation of the functions defined in formula (2.1), are analytic in the set C \ (∓Σ), where C is the space of complex numbers, and Σ is the support of the distribution of the random variable ξ. In particular, K^{±}(r) is

continuous in the points r ∈ A^{±}. The numbers r satisfying the equation E \frac{ξ}{(ξ ± r)^{2}} = 0

have no strictly positive condensation points.

Lemma 3.

lim_{r→∞} G(r) = 1
lim_{r→0} G(r) = P (ξ = 0) (= 0 if the distribution of ξ has no atom in 0.)

(2.6)

The second derivative of H(r, φ) with respect to the variable φ is non-positive in the

point φ̄(r), and it can be zero only if either E \frac{ξ}{(r + ξ)^{2}} = 0 (in which case φ̄(r) = 0) or

if E \frac{ξ}{(r − ξ)^{2}} = 0 (in which case φ̄(r) = π). More explicitly,

\frac{∂^{2}}{∂φ^{2}} H(r, φ)|_{φ=φ̄(r)} = −2E \frac{r^{2}ξ^{2} sin^{2} φ}{(r^{2} + ξ^{2} + 2rξ cos φ)^{2}} if 0 < φ̄(r) < π

\frac{∂^{2}}{∂φ^{2}} H(r, φ)|_{φ=φ̄(r)} = −E \frac{rξ}{(r + ξ)^{2}} (= −K^{+}(r)) if φ̄(r) = 0

\frac{∂^{2}}{∂φ^{2}} H(r, φ)|_{φ=φ̄(r)} = E \frac{rξ}{(r − ξ)^{2}} (= −K^{−}(r))  if φ̄(r) = π.

(2.7)

The above relations imply that the saddle point equation (1.4) (together with property (1.5)) has a unique solution for P (ξ = 0) < α(n) < 1, since a pair (r, φ) is a solution if and only if φ = φ̄(r), where φ̄(r) is defined in Lemma 1, and G(r) = α(n).

Let us rewrite formula (1.2) in the form

S^{(k)}(n) = ℜ (\frac{1}{π} ∫_{0}^{π} exp \{Z_{n}(r, φ)\} dφ)  (2.8)

with

Z_{n}(r, φ) = ∑_{j=1}^{n} β_{j}(r, φ)  (2.9)

and

β_{j}(r, φ) = \frac{1}{2} log (r^{2} + ξ_{j}^{2} + 2rξ_{j} cos φ)

+ i arccos \frac{r cos φ + ξ_{j}}{(r^{2} + ξ_{j}^{2} + 2rξ_{j} cos φ)^{1/2}} − α(n)(log r + iφ),

(2.10)

where r is the first coordinate of the solution (r, φ̄) 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

Property B. Let (r, φ̄) = (r(α^{∗}), φ̄(r(α^{∗})) be the solution of the fixed point equation (1.4) (together with relation (1.5)), if α(n) is replaced by α^{∗} = lim_{n→∞} α(n). Then

E \frac{ξ}{(r ± ξ)^{2}} ≠ 0 for r = r(α^{∗}).

The integral in formula (2.8) can be well estimated. To do this we apply a Taylor expansion for β_{j}(r, φ) in the variable φ around the saddle point φ̄ and then sum it up to get a good estimate for Z_{n}(r, φ) defined in (2.9). The coefficients of this Taylor expansion are random. But since the random functions β_{j}(r, φ̄) 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 \frac{∂H(r, φ)}{∂φ} = 0, and the imaginary part

equals

\frac{∂}{∂φ}E arccos \frac{r cos φ + ξ}{(r^{2} + ξ^{2} + 2rξ cos φ)^{1/2}} − α(n) = r \frac{∂H(r, φ)}{∂r} − α(n) = 0  (2.11)

in the point of solution (r, φ̄) 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 β_{j}(r, φ) with respect to the variable φ is non-positive in the asymptotic saddle point (r, φ̄(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 φ̄(r)) with probability almost one (depending on n). In this small neighbourhood of the point φ(r) a small error is committed if all terms β_{j}(r, φ) in (2.8) are replaced by their Taylor expansion around the point φ̄ 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 α^{∗} = lim_{n→∞} α(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 α^{∗} where it does not hold has no condensation points in the open interval (P (ξ = 0), 1). Moreover, in certain cases we know that this set is empty. This is the case for instance if ξ has a symmetric distribution, since under this condition φ̄(r) = π/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, φ) − α 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.

Lemma 4. Let the distribution of ξ satisfy Property A and (2.3). Beside this, let property B be satisfied with r^{∗} = lim_{n→∞} r_{n}, where r_{n} is the solution of the asymptotic saddle point equation (1.4) (together with (1.5)) with the parameter α(n). Let us also assume that the level α^{∗} = lim_{n→∞} α(n) is stable. (This notion will be introduced a bit later.) Put

S̄^{(k)}(n) = \{ \begin{matrix}\begin{matrix}√ & \overline{2} \\ √ & \overline{Knπ} \end{matrix} & exp & \{nA_{0} + \sqrt{n}S_{0} − U_{1}\} & cos & (nB_{0} + T_{0} − U_{2} − \frac{ω}{2} ) \\ if 0 < φ̄(r^{∗}) < π \\ = \frac{1}{\sqrt{2|A_{2}|πn}} exp \{\frac{T_{1}^{2}}{2A_{2}} + nA_{0} + \sqrt{n}S_{0}\} if φ̄(r^{∗}) = 0 \\ = (−1)^{k(n)} \frac{1}{\sqrt{2|A_{2}|πn}} exp \{\frac{T_{1}^{2}}{2A_{2}} + nA_{0} + \sqrt{n}S_{0}\} if φ̄(r^{∗}) = π. \end{matrix} (2.12)

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 ω = ω(n) are defined in (4.3), (4.4) and (4.14′). Then

\frac{S^{(k)}(n)}{S̄^{(k)}(n)} ⇒ 1

where ⇒ denotes convergence in probability.

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 S̄^{(k)}(n) defined in (2.12) when we are interested in its limit behaviour. The expression S̄^{(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} = \{\frac{j}{p}; j = 0, …, p − 1\} with some positive integer p.

A coset of a finite subgroup G_{0} is of the form G_{0} + α with some 0 ≤ α < 1. For all probability measures μ on (0, 1] there is a smallest closed set, called the support of the measure, whose μ measure is one. For all probability measures μ there is a minimal coset G_{0} + α which contains the support of μ. This means that the μ 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 μ. Now we formulate the following

Proposition A. Let (X_{n}, Y_{n}), n = 1, 2, …, 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 Σ, Y_{n} is a random variable on the unit circle G = [0, 1). Let G_{0} + α be the minimal coset which

contains the support of the distribution of Y_{n}. Put U_{n} = \frac{1}{\sqrt{n}} ∑_{s=1}^{n} X_{s}, V_{n} = ∑_{s=1}^{n} Y_{s} − nα.

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 Σ, 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, α can be chosen in an arbitrary way, e.g. α = 0.

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 α(n). This parameter satisfies the relation α(n) → α^{∗}, 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, …, 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.

Definition. We call a probability measure μ on the group G = [0, 1), mod 1 stable

if for all finite cosets K = \{\frac{j}{p} + c, j = 0, …, p − 1\}, with a positive integer p and

0 ≤ c < 1 μ(K) < 1, or in other words, the minimal coset which contains the support of the measure μ is the whole group G.

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.

Proposition B. For all n let (X_{j}(n), Y_{j}(n)), j = 1, 2, …, 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} → 0 holds with a random variable X in R^{k}, EX = 0, which has a covariance matrix Σ, Y_{j}(n) is a random variable on the unit circle [0, 1) with a distribution μ_{n} on [0, 1) such that μ_{n} ⇒ μ, and μ is a stable probability measure on [0, 1), where ⇒ denotes weak convergence of measures. Define

the random variables U_{n} = \frac{1}{\sqrt{n}} ∑_{s=1}^{n} X_{s}(n) and V_{n} = ∑_{s=1}^{n} Y_{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 Σ, V is uniformly distributed on G = [0, 1), and the random variables U and V are independent.

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:

Definition. We call the level α^{∗} stable if one of the following conditions are satisfied.

1.) Either E \frac{±ξ}{(r ± ξ)^{2}} > 0 for r = r(α^{∗}), i.e. either φ̄(α) = 0 or φ̄(α) = π if α is in a

small neighbourhood of α^{∗}. 2.) or 0 < φ̄(α^{∗}) < π, and the distribution of the random variable

Y = \frac{1}{2π} arccos \frac{r cos φ̄ + ξ}{(r^{2} + ξ^{2} + 2rξ cos φ̄)^{1/2}} ,

where r = r(α^{∗}) and φ̄ = φ̄(α^{∗}) is a stable distribution on the unit circle [0, 1).

We can give a good asymptotic of the symmetric statistics S^{(k)}(n) if \frac{n − k}{n} = α(n) → α^{∗} with a stable level α^{∗}.

If 0 < φ̄(α^{∗}) < π, then the second condition of the stability of α^{∗} holds in the generic case, but the description of the exceptional numbers α^{∗} and distributions F seems to be a hard number theoretic problem. Now we formulate the following Theorem.

Theorem 1. Let Property A and relation (2.3) hold, and let α^{∗} be a stable level. If

α(n) = \frac{n − k}{n} → α^{∗} as n → ∞, then the random variables

\frac{log |S^{(k)}(n)| − nA_{0}(n)}{\sqrt{n}}  (2.13)

(with S^{(k)}(n) defined in (1.1)) converge in distribution to the normal law with expecta-

tion zero and variance Var η, where η = η(φ̄) = \frac{1}{2} log (r(α^{∗})^{2} + ξ^{2} + 2r(α^{∗})ξ cos φ̄(α^{∗})), (r(α^{∗}), φ(α^{∗})) is the solution of the saddle-point equation (1.4) if the number α(n) is replaced by α^{∗} = lim_{n→∞} α(n), and A_{0} = A_{0}(n) is defined in (4.3). |S^{(k)}(n)| can be re-

placed by S^{(k)}(n) in the case φ̄(α^{∗}) = 0, by (−1)^{k}S^{(k)}(n) in the case φ̄(α^{∗}) = π in (2.13), while in the case 0 < φ̄(α^{∗}) < π P ( sign S^{(k)}(n) → 1) = 1/2 and log |S^{(k)}(n)| and sign S^{(k)}(n) are asymptotically independent.

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 ξ_{j} have the distribution P (ξ_{j} = 1) = P (ξ_{j} = −1) = 1/2. In this case the random variable η is constant, Var η = 0, and the limit (2.13) is degenerate. In the following Lemma 5 we describe those distributions F and levels α^{∗} for which the limit distribution in Theorem 1 is degenerate. Then we shall describe the limit behaviour of S_{n}^{(k)} in such cases.

Lemma 5. The random variable η = η(α^{∗}) = \frac{1}{2} log (r(α^{∗})^{2} + ξ^{2} + 2rξ cos φ̄(α^{∗})) appearing in Theorem 1 is constant, if an F distributed random variable ξ is concentrated in two points, i.e. there are two numbers x_{1}, x_{2} such that P (ξ = x_{1}) = p, P (ξ = x_{2}) = q = 1 − p, and one of the following conditions is satisfied. a.) 0 < φ̄(α^{∗}) < π, in which case Eξ = px_{1} + qx_{2} = 0, α^{∗} > 1 − 4pq.

b.) φ̄(α^{∗}) = 0, in which case α^{∗} = −\frac{(p − q)(x_{1} + x_{2})}{x_{1} − x_{2}}, Eξ = px_{1} + qx_{2} ≥ 0 and

x_{1} + x_{2} < 0.

c.) φ̄(α^{∗}) = π, in which case α^{∗} = −\frac{(p − q)(x_{1} + x_{2})}{x_{1} − x_{2}}, Eξ = px_{1} + qx_{2} ≤ 0, and

x_{1} + x_{2} > 0.

In Theorem 2 we describe the limit behaviour of S^{(k)}(n) in case a.) of Lemma 5. It contains the result of [4].

Theorem 2. Let the distribution of the random variable ξ have the form P (ξ = x_{1}) = p, P (ξ = x_{2}) = q = 1 − p, px_{1} + qx_{2} = 0, i.e. Eξ = 0. Let \frac{n−k}{n} = α(n) → α^{∗} with some stable level α^{∗} such that 1 > α^{∗} > 1 − 4pq. Then the random variables

\begin{matrix}√ & \overline{Kπn} \\ √ & \overline{2} \end{matrix}e^{−A_{0}(n)}S^{(k)}(n)

converge in distribution to the random variable exp \{\frac{A_{2}(S^{2} − T^{2}) + 2B_{2}ST}{2(A_{2}^{2} + B_{2}^{2})} \} cos Z as

n → ∞, 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 → ∞, (S, T) is a Gaussian random vector with expectation zero, Z is a random variable, uniformly distributed in [0, 2π) and independent of the vector (S, T), and

ES^{2} = Var \frac{−rξ sin φ̄}{r^{2} + ξ^{2} + 2rξ cos φ̄} ET^{2} = Var \frac{rξ cos φ̄ + r^{2}}{r^{2} + ξ^{2} + 2rξ cos φ̄},

Cov (S, T) = Cov (\frac{−rξ sin φ̄}{r^{2} + ξ^{2} + 2rξ cos φ̄}, \frac{rξ cos φ̄ + r^{2}}{r^{2} + ξ^{2} + 2rξ cos φ̄}) ,

(2.14)

where r = r(α^{∗}), φ̄ = φ̄(α^{∗}).

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 −ξ_{j} which satisfy Part b.).

Theorem 2′. Let the distribution of ξ satisfy the following conditions: P (ξ = x_{1}) = p, P (ξ = 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 α^{∗} = \frac{(p − q)(−x_{1} − x_{2})}{x_{1} − x_{2}}. If \frac{n − k}{n} = α(n) → α^{∗}, then the symmetric

polynomial S^{(k)}(n) satisfies the following limit theorem:

\sqrt{2|A_{2}|πn}e^{−nA_{0}(n)}S^{(k)}(n) ⇒ exp \{\frac{T^{2}}{A_{2}} \} if \sqrt{n}(α(n) − α^{∗}) → 0

\sqrt{2|A_{2}|πn}e^{−nA_{0}(n)}S^{(k)}(n) ⇒ exp \{\frac{T^{2}}{A_{2}} + cLV \} if \sqrt{n}(α(n) − α^{∗}) → c, 0 < |c| < ∞,

where L = \frac{\sqrt{pq}(x_{1} − x_{2})}{px_{1} + qx_{2}}, T = −\frac{x_{1} + x_{2}}{x_{1} − x_{2}} V, and V is a standard normal random vari-

able.

If |\sqrt{n}(α(n) − α^{∗})| → ∞, there is not such a natural scaling of S^{(k)}(n) as in the previous cases.

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 ≤ φ ≤ π, which also satisfies relation (1.5).

Proof of Lemma 1. Let us define the function L(r, ψ) = \frac{1}{2}E log(r^{2} + ξ^{2} + 2rξψ), −1 ≤ ψ ≤ 1. This function is obtained if ψ is written instead of cos φ in the function H(r, φ). It is a concave function of the variable ψ in the open interval −1 < ψ < 1 for all r > 0, since its second derivative is negative. The behaviour of the function L(r, ψ) in the end point ψ = 1 can be investigated by means of the following observation. There is a sufficiently small ε > 0 such that in the interval 1−ε < ψ < 1 either L(r, ψ) is monotone

decreasing and the derivative \frac{∂L(r, ψ)}{∂ψ} is negative or L(r, ψ) is monotone increasing and

the derivative \frac{∂L(r, ψ)}{∂ψ} is positive. In the first case

L(r, 1)−L(r, ψ) = \frac{1}{2}E log (1 + \frac{2(1 − ψ)rξ}{r^{2} + ξ^{2} + 2rξψ} ) ≤ E \frac{(1 − ψ)rξ}{r^{2} + ξ^{2} + 2rξψ} = (1−ψ)\frac{∂L}{∂ψ} < 0,

and L(r, 1) < sup L(r, ψ).

In the second case it follows from formula (2.3) and Fatou’s lemma that

0 ≤ lim sup_{ψ→1} \frac{∂L}{∂ψ} = lim sup_{ψ→1} E \frac{rξ}{r^{2} + ξ^{2} + 2rξψ} ≤ E \frac{rξ}{(r + ξ)^{2}} = rK^{+}(r),

where the function K^{+}(r) is defined in (2.1). Hence r ∈ A^{+}, and Property A can be applied. This implies in particular that L(r, 1) = lim_{ψ→1} L(r, ψ) = sup_{0≤ψ≤1} L(r, ψ).

Similarly, ψ = −1 is the maximum of L(r, ψ) if and only if the function L(r, ψ) is monotone decreasing in the interval (−1, −1 + ε) with a sufficiently small ε > 0, and r ∈ A^{−}, i.e. K^{−}(r) ≥ 0. In particular, the function L(r, ψ) is continuous in the point ψ = −1 in this case.

The above results imply that the function H(r, φ) has a unique maximum in the interval 0 ≤ φ ≤ π. The maximum is in the point φ = 0 if the function L(r, ψ) has its maximum at ψ = 1 which holds if K^{+}(r) ≥ 0. It has its maximum at φ = π if L(r, ψ) has its maximum at ψ = −1 and K^{−}(r) ≥ 0. These statements are equivalent to the first two lines of formula (2.4). The maximum is in the open interval 0 < φ < π if

−K^{−}(r) < 0 < K^{+}(r). In this case \frac{∂H(r, φ)}{∂φ} = 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 φ̄ = 0 and φ̄ = π. To see

this, observe that since r ∈ A^{−} if φ̄ = 0, r ∈ A^{+} if φ̄ = π, \frac{∂H(r, φ)}{∂φ} = 0 in the place

of maximum φ̄, and the order of differentiation and expectation can be changed in this case too.

Let us introduce the notation U = U(r, ξ, φ) = r^{2} + ξ^{2} + 2rξ cos φ. Now we turn to the

Proof of Lemma A. If z_{0} = r_{0}e^{iφ_{0}} ∉ −Σ, and ξ ∈ Σ then for all z = re^{iφ} in a sufficiently small neighbourhood of z_{0} the number |z + ξ|^{2} = U(r, ξ, φ) ≥ C > 0 with an appropriate number C = C(z_{0}). Hence the function log U(r, ξ, φ) is analytic in such a small neighbourhood of z_{0}, and it is separated from −∞ (independently of ξ ∈ Σ). Then, since log U(r, ξ, φ) ≤ const. (|ξ| + r), and relation (2.3) holds, we get by taking expectation that H(z) = \frac{1}{2}E log U(r, ξ, φ) is analytic in a small neighbourhood of z_{0}.

Similarly, if z_{0} ∉ ∓Σ, ξ ∈ Σ and z is in a small neigbourhood of z_{0}, then

|\frac{ξ}{(ξ ± z)^{2}} | ≤ C < ∞, and taking expectation we get that the functions K^{±}(z) are

analytic in the domain C \ (∓Σ). In particular, Property A implies that the function K^{±}(r) is continuous in the points r ∈ A^{±}.

Moreover, the function K^{±}(r) defined for all r > 0 is upper semicontinuous, hence the sets A^{±} defined in (2.2) are closed subsets of the positive numbers. We show that there is no sequence r_{n}, n = 1, 2, …, with a limit 0 < r = lim_{n→∞} r_{n} < ∞ such that

K^{±}(r_{n}) = 0 for all n. Indeed, the limit r would be also in the set A^{±}, and because of Property A the relation d(r, ∓Σ) > 0 would hold. This would imply that K^{±}(z) ≡ 0 in the domain of analiticity of the function K^{±}(z). This relation also would imply that

E \frac{ξ}{ξ ± z} = 0 on the set ℑz > 0, since the derivative of this function is K^{±}(z) ≡ 0, and

as a consequence it is a constant function. Then choosing z = iu, u → ∞ we get that this constant is zero. On the other hand, we get with the choice z = iu, u → 0 that this constant is P (ξ ≠ 0) ≠ 0, and this is a contradiction.

Now we turn to the

Proof of Lemma 2. We shall prove that

\frac{dG(r)}{dr} > 0 if E \frac{ξ}{(r + ξ)^{2}} < 0 < E \frac{ξ}{(r − ξ)^{2}} (or equivalently, if 0 < φ̄(r) < π), (3.1)

and also

\frac{dG(r)}{dr} > 0  if E \frac{ξ}{(r + ξ)^{2}} > 0 or E \frac{ξ}{(r − ξ)^{2}} < 0.  (3.2)

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 < ∞, \frac{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 \frac{ξ}{r^{2} + ξ^{2} + 2rξ cos φ̄(r)} = 0. This

identity determines the function φ̄(r) in the small neighbourhood of a point (r, φ̄(r)).

The implicit function theorem enables us to calculate the function φ̄′(r). We get that

φ̄′(r) = \frac{E \frac{2ξ(r + ξ cos φ̄)}{U^{2}}}{E \frac{2ξ^{2}r sin φ̄}{U^{2}}} = \frac{cos φ̄}{r sin φ̄} + \frac{E \frac{ξ}{U^{2}}}{sin φ̄E \frac{ξ^{2}}{U^{2}}} .  (3.3)

Exploiting again that the third line of formula (2.4) holds in this case, we get that

G(r) = E(r, φ̄(r)) = E \frac{r^{2} + rξ cos φ̄(r)}{U(r, ξ, φ̄(r))} = E \frac{r^{2}}{U(r, ξ, φ̄(r))} = 1 − E \frac{ξ^{2}}{U(r, ξ, φ̄(r))} (3.4)

and

\begin{aligned}\frac{dG(r)}{dr} & = E \frac{2ξ^{2}(r + ξ cos φ̄(r))}{U^{2}(r, ξ, φ̄(r))} − φ̄′(r)E \frac{2rξ^{3} sin φ̄(r)}{U^{2}(r, ξ, φ̄(r))} \\ & = E \frac{2rξ^{2}}{U^{2}(r, ξ, φ̄(r))} − \frac{E \frac{ξ}{U^{2}(r,ξ,φ̄(r))}E \frac{2rξ^{3}}{U^{2}(r,ξ,φ̄(r))}}{E \frac{ξ^{2}}{U^{2}(r,ξ,φ̄(r))}} , \end{aligned}

if 0 < φ̄(r) < π. Hence relation (3.1) is equivalent to the inequality

E \frac{r^{2}ξ}{U^{2}} E \frac{ξ^{3}}{U^{2}} < (E \frac{rξ^{2}}{U^{2}} )^{2} ,

or since the third line in formula (2.4) implies that

E \frac{rξ^{2}}{U^{2}} = \frac{1}{2 cos φ̄}E \frac{2rξ^{2} cos φ̄ − ξU}{U^{2}} = −\frac{1}{2 cos φ̄} (E \frac{ξ^{3}}{U^{2}} + E \frac{r^{2}ξ}{U^{2}} )

it is also equivalent to the inequality

(4 cos^{2} φ̄ − 2)E \frac{ξ^{3}}{U^{2}} E \frac{r^{2}ξ}{U^{2}} < (E \frac{ξ^{3}}{U^{2}} )^{2} + (E \frac{r^{2}ξ}{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} φ̄− 2| < 2 for 0 < φ̄ < π, and the equations E \frac{ξ^{3}}{U^{2}} = 0 and E \frac{r^{2}ξ}{U^{2}} = 0 cannot hold simultaneously. Indeed, they would imply together with the third line of formula

(1.4) for r > 0 and 0 < φ̄ < π that E \frac{ξ^{2}}{U^{2}} = 0, and this is impossible.

To prove relation (3.2) let us observe that if E \frac{ξ}{(r + ξ)^{2}} > 0, then φ̄(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 \frac{dG(r)}{dr} are calculated. Simple calculation

shows that G(r) = E(r, φ̄(r)) = E \frac{r}{ξ + r}, \frac{dG(r)}{dr} = E \frac{ξ}{(r + ξ)^{2}} > 0, and if E \frac{ξ}{(r − ξ)^{2}} < 0,

then φ̄(r) = π, and \frac{dG(r)}{dr} = −E \frac{ξ}{(r − ξ)^{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 \frac{ξ}{(ξ ± r)^{2}} = 0. To prove the continuity in these points it is enough

to show that the function φ̄(r) defined in Lemma 1 is continuous in these points. To prove this observe that in these points either φ̄(r) = 0 or φ̄(r) = π. If φ̄(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 < φ ≤ π. This function is uniformly continuous (analytic) and separated from zero in a small neighbourhood of the set {z: z = re^{iφ}}, with this r and ε ≤ φ ≤ π for arbitrary ε > 0. This implies that φ̄(r) is continuous in this exceptional set if φ̄ = 0. The case φ̄ = π can be handled similarly. Lemma 2 is proved.

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} → ∞ and r_{n} → 0. To prove the first relation let us first consider the case when there is a sequence of numbers r_{n} → ∞

such that 0 < φ̄(r_{n}) < π. By relation (3.4), Fatou’s lemma and the observation \frac{r_{n}^{2}}{U} → 1,

\frac{ξ^{2}}{U} → 0, \frac{ξ^{2}}{U} ≥ 0, \frac{r^{2}}{U} ≥ 0 imply that

lim inf_{r→∞} G(r) = lim inf_{r→∞} E \frac{r^{2}}{U} ≥ 1,
lim sup_{r→∞} G(r) = 1 − lim inf_{r→∞} E \frac{ξ^{2}}{U} ≤ 1,

hence the first line of relation of (2.6) holds in this case. Similarly if r_{n} → 0, 0 <

φ̄(r_{n}) < π, then \frac{r^{2}}{U} → I(ξ = 0) and \frac{ξ^{2}}{U} → 1 − I(ξ = 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 φ̄(r) either

φ̄(r) = 0 and E \frac{ξ}{(ξ + r)^{2}} ≥ 0 or φ̄(r) = π and E \frac{−ξ}{(ξ − r)^{2}} ≥ 0 for all r ≥ r_{0} with some

r_{0} > 1 if the case r → ∞ is considered. We claim that −Σ ∩ {r: r > r_{0}} is empty for r > r_{0} in the first case, and Σ ∩ {r: r > r_{0}} is empty for r > r_{0} is empty in the second case, where Σ denotes the support of the distribution of the ξ. 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}) ≥ K2^{−n} with some appropriate K > 0 for all n = 1, 2, …, where F is the distribution function of the random variable ξ. Let R be the intersection of the intervals [a_{n}, b_{n}], n = 1, 2, . . .. Then

R > r_{0}, and we claim that E \frac{ξ}{(ξ + R)^{2}} = −∞ which is a contradiction. This equation holds, because for all n > 0

E \frac{ξ}{(ξ + R)^{2}} ≤ ∫_{−b_{n}}^{−a_{n}} \frac{x}{(R + x)^{2}} F ( dx) + EξI(ξ ≥ 0) ≤ −const. 2^{n} + const. , (3.5)

and we get the above relation as n → ∞. The proof in the case φ̄ = π for r ≥ r_{0} is similar.

It follows from the above proved statement, the relation lim_{r→∞} \frac{r}{r ± ξ} = 1 with prob-

ability one and Lebesgue convergence theorem that lim_{r→∞} G(r) = lim_{r→∞} E \frac{r}{r ± ξ} = 1 in

this case too.

The limit behaviour in the case r → 0 can be handled similarly. If there is no sequence r_{n} → 0 such that 0 < φ̄(r_{n}) < π, then there is a number 1 > r_{0} > 0 such that either φ̄(r) = 0 or φ̄(r) = π for all 0 < r < r_{0}. In the first case −Σ∩{r: 0 < r < r_{0}} = ∅, and in the second case Σ ∩ {r: 0 < r < r_{0}} = ∅. This can be proved similarly to the case r → ∞ with an estimate similar to (3.5) with the difference that in this case the

relation E \frac{ξ}{(R + ξ)^{2}} I(ξ > 0) ≤ E \frac{ξ}{ξ^{2}} I(ξ > 0) ≤ E \frac{I(ξ ≠ 0)}{|ξ|} < ∞ holds.

Finally, as lim_{r→0} \frac{r}{r ± ξ} = I(ξ = 0), the Lebesgue dominated convergence theorem

implies that lim_{r→0} G(r) = lim_{r→0} E \frac{r}{r ± ξ} = EI(ξ = 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 (ξ = 0) < α(n) < 1. Let us calculate the second partial derivative of F (r, φ̄) with respect of the variable φ in the saddle point. We get that

\frac{∂^{2}}{∂φ^{2}} H(r, φ) = −E \frac{rξ cos φ}{U} − 2E \frac{r^{2}ξ^{2} sin^{2} φ}{U^{2}},

in a general point (r, φ). Then a simple substitution implies formula (2.7). Lemma 3 is proved.

4. Asymptotic approximation for the symmetric polynomial S^{(k)}(n).

Let us consider the solution (r_{n}, φ̄_{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 α(n) at the right-hand side of formula (1.4). On the other hand, if (r(α^{∗}), φ̄(α^{∗})) denotes the solution of the equation (1.4) with the modification that the number α(n) is replaced by α^{∗} = lim_{n→∞} α(n) in it, then lim_{n→∞} r_{n} = r(α^{∗}), and lim_{n→∞} φ̄_{n} = φ̄(α^{∗}). Indeed, it follows from Lemma 2 that lim_{n→∞} r_{n} = r(α^{∗}), since the function G(r) which was so defined that the number r_{n} is the solution of the equation G(r) = α(n) is a continuous and strictly monotone function. Then it follows from Lemma 1 that the relation lim_{n→∞} φ̄_{n} = φ̄(α^{∗}) also holds.

We want to make a Taylor expansion of the function β_{j}(r_{n}, φ) defined in formula (2.10) in the variable φ around the point (r_{n}, φ̄_{n}). For this end we introduce some notations. Put

η_{j}^{(0)} = η_{j}^{(0)}(n) = ℜ (β_{j}(r_{n}, φ̄_{n}) − Eβ_{j}(r_{n}, φ̄_{n})) = \frac{1}{2} log (r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j} cos φ̄_{n})

− \frac{1}{2}E log (r_{n}^{2} + ξ^{2} + 2r_{n}ξ cos φ̄_{n}) ,

ζ_{j}^{(0)} = ζ_{j}^{(0)}(n) = ℑ (β_{j}(r_{n}, φ̄_{n}) − Eβ_{j}(r_{n}, φ̄_{n})) = arccos \frac{r_{n} cos φ̄_{n} + ξ_{j}}{(r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j} cos φ̄_{n})^{1/2}}

− E arccos \frac{r_{n} cos φ̄_{n} + ξ}{(r_{n}^{2} + ξ^{2} + 2r_{n}ξ cos φ̄_{n})^{1/2}} ,

(4.1)

η_{j}^{(1)} = η_{j}^{(1)}(n) = \frac{∂}{∂φ}ℜ (β_{j}(r_{n}, φ) − Eβ_{j}(r_{n}, φ))|_{φ=φ̄_{n}} = \frac{∂}{∂φ}ℜβ_{j}(r_{n}, φ)|_{φ=φ̄_{n}}
= −\frac{r_{n}ξ_{j} sin φ̄_{n}}{r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ cos φ̄_{n}} ,  (4.2)
ζ_{j}^{(1)} = ζ_{j}^{(1)}(n) = \frac{∂}{∂φ}ℑ (β_{j}(r_{n}, φ̄_{n}) − Eβ_{j}(r_{n}, φ̄_{n})) = \frac{∂}{∂φ}ℑβ_{j}(r_{n}, φ)|_{φ=φ̄_{n}}

= \frac{r_{n}ξ_{j} cos φ̄_{n} + r_{n}^{2}}{r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j} cos φ̄_{n}} − α(n),

(in the last identity we applied the same calculation as in formula (2.11))

A_{0} = A_{0}(n) = Eℜβ_{j}(r_{n}, φ̄_{n}) = \frac{1}{2}E log (r_{n}^{2} + ξ^{2} + 2r_{n}ξ cos φ̄_{n}) − α(n) log r_{n},
B_{0} = B_{0}(n) − Eℑβ_{j}(r_{n}, φ̄_{n}) = E arccos \frac{r_{n} cos φ̄_{n} + ξ}{(r_{n}^{2} + ξ^{2} + 2r_{n}ξ cos φ̄_{n})^{1/2}} − α(n)φ̄_{n},

(4.3)

the numbers A_{2} = A_{2}(n) and B_{2} = B_{2}(n) which are the second derivatives of the functions Eℜβ_{j}(r_{n}, φ) and Eℑβ_{j}(r_{n}, φ) in the point φ = φ̄_{n}, i.e.

A_{2} = A_{2}(n) = −E \frac{r_{n}ξ cos φ̄_{n}}{U(r_{n}, ξ, φ̄_{n})} − 2E \frac{r_{n}^{2}ξ^{2} sin^{2} φ̄_{n}}{U(r_{n}, ξ, φ̄_{n})^{2}} ,

B_{2} = B_{2}(n) = −E \frac{r_{n}ξ sin φ̄_{n}}{U(r_{n}, ξ, φ̄_{n})} + 2E \frac{r_{n}ξ sin φ̄_{n}(r_{n}ξ cos φ̄_{n} + r^{2})}{U(r_{n}, ξ, φ̄_{n})^{2}},

(4.4)

η_{j}^{(2)} = η_{j}^{(2)}(n) = ℜ\frac{∂^{2}}{∂φ^{2}} (β_{j}(r_{n}, φ) − Eβ_{j}(r_{n}, φ))|_{φ=φ̄_{n}}

= −\frac{r_{n}ξ_{j} cos φ̄_{n}}{U(r_{n}, ξ_{j}, φ̄_{n})} − 2\frac{r_{n}^{2}ξ_{j}^{2} sin^{2} φ̄_{n}}{U(r_{n}, ξ_{j}, φ̄_{n})^{2}} − A_{2},

ζ_{j}^{(2)} = ζ_{j}^{(2)}(n) = ℑ\frac{∂^{2}}{∂φ^{2}} (β_{j}(r_{n}, φ) − Eβ_{j}(r_{n}, φ))|_{φ=φ̄_{n}}

= \frac{−r_{n}ξ_{j} sin φ̄_{n}}{U(r_{n}, ξ_{j}, φ̄_{n})} + 2\frac{r_{n}ξ_{j} sin φ̄_{n}(r_{n}ξ_{j} cos φ̄_{n} + r_{n}^{2})}{U(r_{n}, ξ_{j}, φ̄_{n})^{2}} − B_{2}.

(4.5)

We can write

ℜβ_{j}(r_{n}, φ) = A_{0} + η_{j}^{(0)}(r_{n}, φ̄_{n}) + η_{j}^{(1)}(r_{n}, φ̄_{n})(φ − φ̄_{n})

+ \frac{1}{2} (A_{2} + η_{j}^{(2)}) (φ − φ̄_{n})^{2} + \frac{1}{6}ϑ_{j,1}(φ − φ̄_{n})^{3},

ℑβ_{j}(r_{n}, φ) = B_{0} + ζ_{j}^{(0)}(r_{n}, φ̄_{n}) + ζ_{j}^{(1)}(r_{n}, φ̄_{n})(φ − φ̄_{n})

+ \frac{1}{2} (B_{2} + ζ_{j}^{(2)}) (φ − φ̄_{n})^{2} + \frac{1}{6}ϑ_{j,2}(φ − φ̄_{n})^{3},

where

\begin{aligned}ϑ_{j,1} = ϑ_{j,1}(r_{n}, φ) & = \frac{∂^{3}}{∂φ^{3}} ℜβ_{j}(r_{n}, φ)|_{φ=φ̃} , \\ ϑ_{j,2} = ϑ_{j,2}(r_{n}, φ) & = \frac{∂^{3}}{∂φ^{3}} ℑβ_{j}(r, φ)|_{φ=˜φ̃} \end{aligned}  (4.6)

with some numbers φ̃ and ˜ φ̃ in the interval [φ, φ̄_{n}]. Summing up the last relations for j = 1, …, n, we get the following relation for the function Z_{n}(r_{n}, φ) defined in formula (2.9):

\begin{aligned}Z_{n}(r, φ) & = n (A_{0} + iB_{0}) + \sqrt{n}S_{0}(n) + iT_{0}(n) + \sqrt{n}(S_{1}(n) + iT_{1}(n))(φ − φ̄_{n}) \\ & + \frac{n}{2}(A_{2} + iB_{2})(φ − φ̄_{n})^{2} + \frac{\sqrt{n}}{2}(ε_{1}(n) + iε_{2}(n))(φ − φ̄_{n})^{2} \\ & + \frac{n}{6}(δ_{1}(n) + iδ_{2}(n))(φ − φ̄_{n})^{3}, \end{aligned} (4.7)

where

S_{0} = S_{0}(n) = \frac{1}{\sqrt{n}} ∑_{j=1}^{n} η_{j}^{(0)}(r_{n}, φ̄_{n}) and T_{0} = T_{0}(n) = ∑_{j=1}^{n} ζ_{j}^{(0)}(r_{n}, φ̄_{n}) mod 2π,

(4.8)

S_{1} = S_{1}(n) = \frac{1}{\sqrt{n}} ∑_{j=1}^{n} η_{j}^{(1)}(r_{n}, φ̄_{n}),  T_{1} = T_{1}(n) = \frac{1}{\sqrt{n}} ∑_{j=1}^{n} ζ_{j}^{(1)}(r_{n}, φ̄_{n}), (4.9)

and

ε_{1}(n) = \frac{1}{\sqrt{n}} ∑_{j=1}^{n} η_{j}^{(2)}(r_{n}, φ̄_{n}),  ε_{2}(n) = \frac{1}{\sqrt{n}} ∑_{j=1}^{n} ζ_{j}^{(2)}(r_{n}, φ̄_{n})

δ_{k}(n) = \frac{1}{n} ∑_{j=1}^{n} ϑ_{j,k}(r_{n}, φ), k = 1, 2.

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

Ī(n) = [φ̄_{n} − n^{−1/2+1/10}, φ̄_{n} + n^{−1/2+1/10}] and I(n) = Ī(n) ∩ [0, π).

Observe that for sufficiently large n Ī(n) = I(n) if 0 < φ̄(α^{∗}) < π, and Ī(n) = I(n) ∪ (−I(n)) if φ̄(α^{∗}) = 0 or φ̄(α^{∗}) = π with α^{∗} = lim_{n→∞} α(n). This relation follows from Lemma 1, the relation lim_{n→∞} φ̄_{n} = φ̄(α^{∗}) which we pointed out at the beginning of this

Section, Property B and the observation that in the case φ̄(α^{∗}) = 0 or π K^{±}(r(α^{∗})) > 0 with a strict inequality. Indeed, the inequality K^{±}(r(α(n))) > 0 also holds in this case. These facts imply the relation between the intervals I(n) and Ī(n) formulated in this paragraph.

We claim that there is an appropriate set Ω(n) on the probability space where the random variables ξ_{1}, ξ_{2}, … are defined such that

P (Ω(n)) → 1 as n → ∞,  (4.10)

and

ℜ (\frac{1}{π} ∫_{I(n)} exp\{Z_{n}(r_{n}, φ)\} dφ) = \{ \begin{matrix}2D_{n} & if 0 < φ̄_{n} < π \\ D_{n} & if φ̄_{n} = 0 or φ̄ = π \end{matrix}  (4.11)

on the set Ω(n) for the function Z_{n}(r, φ) defined in formula (2.9) with a (random) number D_{n} which satisfies the relation

D_{n} = ℜ (\begin{matrix}\sqrt{π} & exp & \{Z_{n}(r_{n}, φ̄_{n}) − \frac{(S_{1}(n) + iT_{1}(n))^{2}}{2(A_{2} + iB_{2})} + O (n^{−1/10})\} \\ √ & \overline{2n(−A_{2} − iB_{2})} \end{matrix}) ,  (4.11′)

where S_{1}(n) and T_{1}(n) are defined in (4.9), A_{0}, B_{0} in (4.3), φ̄_{n} = φ̄(α(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}, φ̄_{n}) → (r(α^{∗}), φ̄(α^{∗})) as n → ∞, and Property A and Lemma A can be applied if φ̄(α^{∗}) = 0 or φ̄(α^{∗}) = π. We also claim that

the angle between the complex numbers \begin{matrix}exp & \{Z_{n}(r_{n}, φ̄_{n}) − \frac{(S_{1}(n) + iT_{1}(n))^{2}}{2(A_{2} + iB_{2})} \} \\ √ & \overline{2n(−A_{2} − iB_{2})} \end{matrix}
and i = \sqrt{−1} is larger than n^{−1/20},  (4.11′′)

and

|∫_{[0,π]\(I(n))} exp{Z_{n}(r_{n}, φ)} dφ| = O (exp {ℜZ_{n}(r_{n}, φ̄_{n}) − const. n^{1/5}}) , (4.12)

and the O(·) is uniform in (4.11) and (4.12) on the sets Ω(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 Ω(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 Ω(n) implies that

|\frac{(S_{1}(n) + iT_{1}(n))^{2}}{2(A_{2} + iB_{2})} | < const. n^{1/10}

on the set Ω(n). This relation together with formulas (4.11′), (4.11′′) imply that

|D_{n}| ≥ const. |ℜ (\begin{matrix}exp & \{Z_{n}(r_{n}, φ̄_{n}) − \frac{(S_{1}(n) + iT_{1}(n))^{2}}{2(A_{2} + iB_{2})} \} \\ √ & \overline{2n(−A_{2} − iB_{2})} \end{matrix})|

≥ n^{−1/20}const. |\begin{matrix}exp & \{Z_{n}(r_{n}, φ̄_{n}) − \frac{(S_{1}(n) + iT_{1}(n))^{2}}{2(A_{2} + iB_{2})} \} \\ √ & \overline{2n(−A_{2} − iB_{2})} \end{matrix}|

≥ exp {ℜZ_{n}(r_{n}, φ̄_{n}) − const. n^{1/10}} .

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 < φ̄(α^{∗}) < π and S^{(k)}(n) = D_{n}(1 + o(1)) if φ̄(α^{∗}) = 0 or φ̄(α^{∗}) = π. Hence to prove Lemma 4 it is enough to give a good estimate on D_{n}. We shall consider the cases 0 < φ̄(α^{∗}) < π, φ̄(α^{∗}) = 0 and φ̄(α^{∗}) = π separately. We get with the help of relation (4.11′) and the identity Z_{n}(r_{n}, φ̄_{n}) = n (A_{0} + iB_{0})+\sqrt{n}S_{0}(n)+ iT_{0}(n) that on the set Ω(n)

D_{n} = \begin{matrix}√ & \overline{2} \\ √ & \overline{Knπ} \end{matrix} exp \{nA_{0} + \sqrt{n}S_{0} − U_{1}\} cos (nB_{0} + T_{0} − U_{2} − \frac{ω}{2} ) (1 + O(n^{−1/10}))
if 0 < φ̄(α^{∗}) < π  (4.13)

with

U_{1} = U_{1}(n) = \frac{A_{2}(S_{1}^{2} − T_{1}^{2}) + 2B_{2}S_{1}T_{1}}{2(A_{2}^{2} + B_{2}^{2})} , U_{2} = U_{2}(n) = \frac{−B_{2}(S_{1}^{2} − T_{1}^{2}) + 2A_{2}S_{1}T_{1}}{2(A_{2}^{2} + B_{2}^{2})} ,

(4.14)

and

K = K(n) = (A_{2}^{2} + B_{2}^{2})^{1/2}, ω = ω(n) = arctan \frac{B_{2}}{A_{2}} ,  (4.14′)

because of the relation

\frac{(S_{1} + iT_{1})^{2}}{2(A_{2} + iB_{2})} = U_{1} + iU_{2}.  (4.14′′)

In the case φ̄(α^{∗}) = 0, B_{0} = 0, B_{2} = 0, T_{0} = 0 and S_{1} = 0, hence

D_{n} = \frac{1}{\sqrt{2π|A_{2}|n}} exp \{\frac{T_{1}^{2}}{2A_{2}} + nA_{0} + \sqrt{n}S_{0}\} (1 + O(n^{−1/10})) if φ̄(α^{∗}) = 0, (4.13′)

and in the case φ̄(α^{∗}) = π, nB_{0} = n(−π − α(n)) = k(n)π, T_{0} = 0 and S_{1} = 0. Hence

D_{n} = (−1)^{k(n)} \frac{1}{\sqrt{2π|A_{2}|n}} exp \{\frac{T_{1}^{2}}{2A_{2}} + nA_{0} + \sqrt{n}S_{0}\} (1 + O(n^{−1/10})) if φ̄(α^{∗}) = π.

(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 Ω(n) in the form Ω(n) = Ω_{1}(n)∩Ω_{2}(n). Ω_{1}(n) is the set where the above relations hold:

|S_{1}(n)| < n^{1/20},
|T_{1}(n)| < n^{1/20},
|ε_{k}(n)| < n^{1/10}, k = 1, 2,
|δ̄_{k}(n)| < n^{1/10}, k = 1, 2,
|∑_{j=1}^{n} \frac{ξ_{j} cos(φ̄_{n} ± n^{−4/10})}{r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j} cos(φ̄_{n} ± n^{−4/10})}

− nE \frac{ξ cos(φ̄_{n} ± n^{−4/10})}{r_{n}^{2} + ξ^{2} + 2r_{n}ξ cos(φ̄_{n} ± n^{−4/10})}| < n^{11/20},

(4.15)

where

δ̄_{k}(n) = \frac{1}{n} ∑_{j=1}^{n} ϑ̄_{j,k}, k = 1, 2

with

ϑ̄_{j,1} = sup_{|φ−φ̄_{n}|<n^{−1/2+1/10}} |\frac{1}{2} \frac{∂^{3}}{∂φ^{3}} (log (r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j} cos φ))|

ϑ̄_{j,2} = sup_{|φ−φ̄_{n}|<n^{−1/2+1/10}} |\frac{∂^{3}}{∂φ^{3}} arccos \frac{r_{n} cos φ + ξ}{(r_{n}^{2} + ξ^{2} + 2r_{n}ξ cos φ)^{1/2}} | .

The set Ω_{2}(n) is defined as the set where the above relation holds:

|W (n) − \frac{1}{2}| > n^{−1/20} with W (n) = \frac{1}{π} (nB_{0}(n) + T_{0}(n) − U_{2}(n) − \frac{ω(n)}{2}) mod 1
if 0 < φ̄(α^{∗}) < π,
(4.15′)

where B_{0}, T_{0}, U_{2} and ω are defined in (4.1), (4.2), (4.14) and (4.14′).

The above defined set Ω(n) satisfies relation (4.10), since both Ω_{1}(n) and Ω_{2}(n) satisfy it. It holds for Ω_{1}(n) since the random variables \sqrt{n}S_{1}(n), \sqrt{n}T_{1}(n), \sqrt{n}ε_{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 nδ̄_{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(α^{∗}) = r(α^{∗})e^{iφ̄(α^{∗})} and z_{n} = r_{n}e^{iφ̄_{n}} are separated from the real line if 0 < φ̄(α^{∗}) < π, they are separated from −Σ if φ̄(α^{∗}) = 0, and from Σ if φ̄(α^{∗}) = π. 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 → ∞. 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 → ∞, 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 < φ̄(α^{∗}) < π observe that by (4.7) and the definition of the set Ω(n)

Z_{n}(r_{n}, φ) − Z_{n}(r_{n}, φ̄_{n}) = n(A_{2} + iB_{2})\frac{(φ − φ̄_{n})^{2}}{2} + \sqrt{n}(S_{1}(n) + iT_{1}(n))(φ − φ̄_{n})
+ O (n^{−1/10})

= \frac{n(A_{2} + iB_{2})}{2} (φ − φ̄_{n} + \frac{S_{1} + iT_{1}}{\sqrt{n}(A_{2} + iB_{2})})^{2} − \frac{(S_{1} + iT_{1})^{2}}{2(A_{2} + iB_{2})}

+ O (n^{−1/10})

(4.16)

if φ ∈ I(n) and ω ∈ Ω(n), hence

∫_{I(n)} exp{Z_{n}(r_{n}, φ) − Z_{n}(r_{n}, φ̄_{n})} dφ

= ∫_{I(n)} exp\{\frac{n(A_{2} + iB_{2})}{2} (φ − φ̄_{n} + \frac{S_{1} + iT_{1}}{\sqrt{n}(A_{2} + iB_{2})})^{2}

− \frac{(S_{1} + iT_{1})^{2}}{2(A_{2} + iB_{2})} + O (n^{−1/10})\} dφ

= ∫_{−∞}^{∞} exp\{\frac{n(A_{2} + iB_{2})}{2} (φ − φ̄_{n} + \frac{S_{1} + iT_{1}}{\sqrt{n}(A_{2} + iB_{2})})^{2}

− \frac{(S_{1} + iT_{1})^{2}}{2(A_{2} + iB_{2})} + O (n^{−1/10})\} dφ + O (e^{−Kn^{1/5}})

= \begin{matrix}\sqrt{2π} & exp & \{−\frac{(S_{1} + iT_{1})^{2}}{2(A_{2} + iB_{2})} + O (n^{−1/10})\} \\ √ & \overline{(−A_{2} − iB_{2})n} \end{matrix},

since ∫_{∞}^{∞} e^{−A(φ−B)^{2}} dφ = \sqrt{\frac{π}{A}} if ℜA > 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 φ̄(α^{∗}) = 0 or φ̄(α^{∗}) = π are similar, but simpler. The integrals we are interested in can be calculated similarly, only the approximating integrals ∫_{−∞}^{∞} must be replaced by ∫_{0}^{∞} or ∫_{−∞}^{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 π in this case.

To prove (4.12) it is enough to show that

ℜZ_{n}(r_{n}, φ) ≤ ℜZ_{n}(r_{n}, φ̄_{n}) − const. n^{1/5} if φ ∈ [0, π] \ I(n)  (4.17)

on the set Ω(n), where the function Z_{n}(r_{n}, φ) is defined in formulas (2.9) and (2.10). First we show the following weaker result:

ℜZ_{n} (r_{n}, φ̄_{n} ± n^{−2/5}) < ℜZ_{n} (r_{n}, φ̄_{n}) − const. n^{1/5},  (4.18)

i.e. relation (4.17) holds if some very special points of the set [0, π]\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 < φ̄ < π in which case

A_{2} = −2E \frac{r_{n}^{2}ξ^{2} sin^{2} φ̄_{n}}{U(r_{n}, ξ, φ̄_{n})^{2}} < −K because of Lemma 1 or φ̄_{n} = 0 or φ̄_{n} = π, and in these

cases A_{2} = E \frac{∓r_{n}ξ}{(r_{n} ± ξ)^{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 φ = φ̄_{n}±n^{−2/5} with the help of

the following observations: nA_{2} \frac{(φ − φ̄_{n})^{2}}{2} < −const. n^{1/5}, \sqrt{n}|(φ − φ̄_{n})S_{1}(n)| < n^{3/20} on the set Ω(n) because of the relation A_{2} < −K and formula (4.15).

Relation (4.17) can be rewritten, with the change of variable ψ = cos φ, in the equivalent form

Y_{n}(ψ) ≤ Y_{n}(cos φ̄_{n}) − const. n^{1/5} if | arccos ψ − φ̄_{n}| ≥ n^{−2/5},  (4.19)

on the set Ω(n), with the function Y_{n}(ψ) defined as

Y_{n}(ψ) = ℜZ_{n}(r_{n}, arccos ψ) = ∑_{j=1}^{n} \frac{1}{2} log (r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j}ψ) − nα(n) log r_{n}.

Relation (4.18) implies that

Y_{n} (cos(φ̄_{n} ± n^{−2/5})) ≤ Y_{n}(cos φ̄_{n}) − const. n^{1/5}.  (4.20)

To prove (4.19) it is enough to observe that

\frac{d^{2}}{dψ^{2}} Y_{n}(ψ) = − ∑_{j=1}^{n} \frac{2r_{n}^{2}ξ_{j}^{2}}{(r_{n}^{2} + ξ_{j}^{2} + 2r_{n}ξ_{j}ψ)^{2}} ≤ 0,  (4.21)

hence the function Y_{n}(·) is concave, and relation (4.20) implies its strengthened form, relation (4.19).

5. Proof of the limit theorems for sums of independent vectors.

Proof of Proposition A. In the proof we apply a natural adaptation of the characteristic function technique. We shall investigate the expressions

φ(t, l) = E exp{itX_{1} + 2πil(Y_{1} − α)},  (5.1)

where t ∈ R^{k}, l is an arbitrary integer if G_{0} = G, l is an integer, 0 ≤ l < p if

G_{0} = \{\frac{j}{p}, j = 0, …, p − 1\}, and tX_{s} denotes scalar product. We claim that

φ(t, l) = exp \{−\frac{1}{2}tΣt^{∗} + o(t^{2})\} if l = 0 and t → 0
|φ(t, l)| < C < 1 if l ≠ 0 and |t| < ε,

(5.2)

where the constants C < 1 and ε > 0 may depend on l,

Since EX_{1} = 0, and the coefficient of Y_{1} − α in the definition of the function φ(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 μ is the whole group G. We show that in this case for all positive integers l and 0 ≤ α ≤ 1 there is some δ = δ(l) > 0 and η = η(l) > 0 depending only on l such that the distribution μ of Y_{s} satisfies the inequality

μ ( ⋃_{j=1}^{l} [\frac{j}{l} − η + α, \frac{j}{l} + η + α]) < 1 − δ.  (5.3)

Let us emphasize that the numbers η > 0 and δ > 0 in formula (5.3) may depend on l but not on α.

To prove (5.3) first we show that for all sets

A(β) = A(β, l, η) = ⋃_{j=1}^{l} (\frac{j}{l} + β − 2η, \frac{j}{l} + β + 2η) ,

μ(A(β)) < 1 − δ if the numbers η = η(β, l) and δ = δ(β, l) are appropriately chosen.

Indeed, the μ measure of the (finite) sets ⋃_{j=1}^{l} \{\frac{j}{l} + β\} is less than one for all 0 ≤

β ≤ 1, since otherwise the support of the measure μ 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 Ā(β), which sets are defined in the same way as A(β), (μ(A(β) < 1 − δ(β)), only 2η is replaced by η in their definition. If we choose η as the minimum of the numbers η appearing in the definition of the sets Ā(β) 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(β) for which Ā(β) takes part in this cover. Hence relation (5.3) holds if η and δ are chosen as the minimum of those values η(β) > 0 and δ(β) > 0 which appear in the sets A(β) for which Ā(β) takes part in this finite cover of G.

Relation (5.3) implies that for sufficiently small ε = ε(l) > 0

P (|lY_{1} + tX_{1} − α| > \frac{η}{2} ) > \frac{δ}{2}

for all 0 ≤ α ≤ 1 if t < ε with some η = η(l) and δ = δ(l). Hence

Eℜe^{i(lY_{1}+tX_{1}−α)} < 1 − \frac{ηδ^{2}}{8}

for all α ∈ [0, 1]. Since this relation holds for all α, this implies the second line of (5.2) in the case G = G_{0}.

If G_{0} + α with G_{0} = \{\frac{j}{p}, j = 0, 1, …, p − 1\}, is the minimal coset containing the

support of μ, then the distribution of Y_{1} − α is concentrated on G_{0}. The distribution

of Y_{1} − α is concentrated on some points of the form \frac{k_{u}}{p}, u = 0, …, r with some r such

that the μ measure of all these points \frac{k_{u}}{p} is positive. We may assume by replacing α

by α − \frac{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}, …, k_{r} and p equals 1. Hence there are some integers N_{u}, u = 1, …, r and N such that

Np + ∑_{u=1}^{r} N_{u}k_{u} = 1.  (5.4)

This fact implies that for any 1 ≤ l < p all vectors exp \{2πil \frac{k_{u}}{p} \}, u = 0, …, r cannot

be parallel. Indeed, otherwise the relation lk_{u} = lk_{0} = 0 mod p would hold for all u = 1, …, r, and this contradicts to (5.4). Also the maximum between the angles of the vectors exp \{itX_{1} + 2π \frac{ilk_{u}}{p} \} 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} + 2πilV_{n}\} = [φ (\frac{t}{\sqrt{n}}X_{1}, l)]^{n}, relation (5.2) implies that

E exp\{itU_{n} + 2πilV_{n}\} = \{ \begin{matrix}exp \{−\frac{1}{2}tΣt^{∗}\} (1 + o(1)) & if l = 0 \\ o(1) & if l ≠ 0, \end{matrix}

Here t ∈ R^{k}, l = 0, ±1, ±2, … if G_{0} = G, and l is an integer, 0 ≤ l < p, if G_{0} =

\{\frac{j}{p}, j = 0, …, p − 1\}. This means that lim_{n→∞} E exp\{itU_{n} + 2πilV_{n}\} = E exp\{itU +

2πilV } 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.

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 φ(t, l) is replaced by φ_{n}(t, l) = E exp{itX_{1}(n)+2πil(Y_{1}(n)−α)}. 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 → 0. The second line of (5.2) can be deduced from a modified version of formula (5.3) where the distribution μ of Y_{1} is replaced by the distribution μ_{n} of Y_{1}(n), but the numbers η and δ must not depend on n. This can be deduced, just as it was done in the proof of Proposition A, from the weaker relation μ_{n}(A(β)) < 1 − δ with

A(β) = A(β, l, η) = ⋃_{j=1}^{l} (\frac{j}{l} + β − 2η, \frac{j}{l} + β + 2η) ,

δ > 0, η > 0, if the numbers η = η(β, l) and δ = δ(β, l) are appropriately chosen. We have already proved in Proposition A that μ(A(β)) < 1 − δ, where μ is the (weak) limit of the measures μ_{n}. Moreover, this statements also holds for the closure Ā(β) of the set A(β) with a possibly smaller parameter η. Since μ_{n} ⇒ μ, lim sup_{n→∞} μ_{n}(Ā(β)) ≤ μ(Ā(β)).

This implies that also the relation μ_{n}(A(β)) < 1 − δ holds for large n. Proposition B is proved.

6. The proof of the main results.

Proof of Theorem 1. By Lemma 4 log |S^{(k)}(n)| − log |S̄^{(k)}(n)| ⇒ 0, where S̄^{(k)}(n) is defined in (2.12), and ⇒ denotes stochastic convergence. Hence S^{(k)}(n) can be replaced by S̄^{(k)}(n) in the proof of Theorem 1.

We claim that

\frac{U_{1}(n)}{\sqrt{n}} ⇒ 0, \frac{T_{1}^{2}(n)}{\sqrt{n}} ⇒ 0 and

\frac{1}{\sqrt{n}} log |cos (nB_{0}(n) + T_{0}(n) − U_{2}(n) − \frac{ω(n)}{2})| ⇒ 0.

(6.1)

The third relation in (6.1) is needed only in the case when 0 < φ(α^{∗}) < π. 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 2π converge in distribution to the uniform distribution in [0, 2π). 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, 2π). Hence the random variables T_{0}(n) − U_{2}(n) mod 2π converge in distribution to the uniform distribution of [0, 2π), as we claimed. This rela-

tion implies that the random variables log |cos (n(B_{0}(n) − U_{2}(n) − \frac{ω(n)}{2})| converge in

distribution to a random variable log | cos V |, where V is uniformly distributed in [0, 2π). 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 η, and a slight refinement of the previous argument also shows that the vectors

(log cos (n(B_{0}(n) − U_{2}(n) − \frac{ω(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, 2π], and Z is normally distributed with expectation zero and variance Var η. Relation (2.13) follows from the above observations. Because of Lemma 4, the form of S̄^{(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.

Proof of Lemma 5. The random variable η = η(α^{∗}) is constant if and only if

ξ^{2} + 2r(α^{∗})ξ cos φ̄(α^{∗}) = const. with probability 1.

Since ξ 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(α^{∗})x_{1} cos φ̄(α^{∗}) = x_{2}^{2} + 2r(α^{∗})x_{2} cos φ̄(α^{∗}), or equivalently x_{1} + x_{2} + 2r(α^{∗}) cos φ̄(α^{∗}) = 0. In case a.) when the relation 0 < φ̄(α^{∗}) < π holds, by Lemma 1 the identity E \frac{ξ}{r^{2}(α^{∗}) + ξ^{2} + 2r(α^{∗})ξ cos φ̄(α^{∗})} = 0 must hold. This is equivalent to

the relation px_{1} + qx_{2} = 0 with p = P (ξ = x_{1}), q = P (ξ = x_{2}) = 1 − p, since r^{2}+x_{1}^{2}+2rx_{1} cos φ̄ = r^{2}+x_{2}^{2}+2rx_{2} cos φ̄ in this case. Finally, the second equation of the

fixed point equation (1.4) r \frac{∂H}{∂r} |_{r=r(α^{∗})} = α^{∗} yields that E \frac{rξ cos φ̄ + r^{2}}{r^{2} + ξ^{2} + 2rξ cos φ̄} = α^{∗}.

This is equivalent to \frac{r^{2}}{r^{2} − x_{1}x_{2}} = α^{∗}, since in this case r^{2}+ξ^{2}+2rξ cos φ̄ = r^{2}−x_{1}x_{2}, as

the calculation r^{2} +ξ^{2} +2rξ cos φ̄ = r^{2} +(px_{1}^{2} +qx_{2}^{2}) = r^{2} +(px_{1} +qx_{2})(x_{1} +x_{2})−x_{1}x_{2} = r^{2} − x_{1}x_{2} shows.

We have proved that the distribution of the random variable ξ must be concentrated in two different points, and the above equations make possible to calculate r(α^{∗}) and φ̄(α^{∗}) from α^{∗}. To decide whether we get a real solution for a pair (F, α^{∗}) we have to check whether the condition | cos(φ(α^{∗})| < 1 is satisfied. Some calculation shows that

cos φ̄(α^{∗}) = −\frac{x_{1} + x_{2}}{2r(α^{∗})} = −\frac{(q − p)x_{1}}{2qr(α^{∗})} , r(α^{∗})^{2} = \frac{p}{q} \frac{α^{∗}}{1 − α^{∗}} x_{1}^{2}. The last two identities

yield that cos^{2} φ̄(α^{∗}) = \frac{(p − q)^{2}}{4pq} \frac{1 − α^{∗}}{α^{∗}}. This gives that the condition | cos φ̄(α^{∗})| < 1

is equivalent to α^{∗} > 1 − 4pq.

In case b.) when the relation φ̄(α^{∗}) = 0 holds the random variable ξ is concentrated

in two points x_{1}, x_{2}, x_{1} + x_{2} + 2r(α^{∗}) = 0, and E \frac{ξ}{(r + ξ)^{2}} ≥ 0. The latter relation is

equivalent to Eξ ≥ 0 in the present case. Since 2r(α^{∗}) = −(x_{1} + x_{2}) the second part

of the fixed point equation (1.4) yields that α^{∗} = E \frac{r}{r + ξ} = −\frac{(p − q)(x_{1} + x_{2})}{x_{1} − x_{2}}. The

conditions px_{1} + qx_{2} ≥ 0, x_{1} + x_{2} < 0 are satisfied. The last condition appears, because it is equivalent to r(α^{∗}) > 0. Some calculation shows that under such conditions the relation 0 < α^{∗} < 1 also holds. Case c.) in Lemma 5 when φ̄(α^{∗}) = π can be handled similarly to case b.). Lemma 5 is proved.

Proof of Theorem 2. Because of Lemma 4 the random variable S^{(k)}(n) can be replaced by S̄^{(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} − \frac{ω}{2} mod 2π) .  (6.2)

converge in distribution to a random vector (U, Z), where Z = Z_{1}−U_{2}+const. mod 2π

with U_{2} = \frac{−B_{2}(S^{2} − T^{2}) + 2A_{2}ST}{2(A_{2} + B_{2}^{2})} , U_{1} = −U = −\frac{A_{2}(S^{2} − T^{2}) + 2B_{2}ST}{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, 2π), and it is independent of the vector (S, T). These relations imply that the random variable Z is also uniformly distributed in [0, 2π), 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, 2π) 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.

Proof of Theorem 2′. Here again the investigation of the random variable S^{(k)}(n) can be replaced by that of S̄^{(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) = ∑_{j=1}^{n}(η_{j}^{(0)} − Eη_{j}^{(0)}) with η_{j}^{(0)} = log |r(α(n)) + ξ_{j}|. Under the

conditions of Theorem 2′ lim_{n→∞} Var η_{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(α(n)) = r(α^{∗}) + (α(n) − α^{∗})r′(α^{∗}) + O ((α(n) − α^{∗})^{2}), and since x_{1} + x_{2} + 2r(α^{∗}) = 0, η_{j} ∼

log |ξ_{j} − \frac{x_{1} + x_{2}}{2} + r′(α^{∗})(α(n) − α^{∗})|. Hence η_{j} takes two values y_{1} and y_{2} with prob-

abilities p and q, and |y_{1} − y_{2}| = \frac{4r′(α^{∗})|α(n) − α^{∗}|}{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

φ̄(α) = 0 in a small neighbourhood of α^{∗} that r′(α^{∗})E \frac{ξ}{(r + ξ)^{2}} = 1. Because of this

identity and the relation x_{1} + x_{2} + 2r(α^{∗}) = 0 that r′(α^{∗}) = \frac{(x_{1} − x_{2})^{2}}{4(px_{1} + qx_{2})}. Hence

VarS_{0}(n) = Var η_{j} = pq(y_{1} − y_{2})^{2} ∼ pq(α(n) − α^{∗})^{2} \frac{(x_{1} − x_{2})^{2}}{(px_{1} + qx_{2})^{2}} . On the other hand,

some calculation yields that Var T_{1}(n) = \frac{(x_{1} + x_{2})^{2}}{(x_{1} − x_{2})^{2}} . Since the random variables ξ_{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, \frac{x_{1} + x_{2}}{x_{1} − x_{2}} V ) with the

choice L_{n} = \sqrt{pq}|α(n) − α^{∗}|\frac{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′.

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.

References:

1.) Dynkin, E. B., Mandelbaum, A. (1983) Symmetric statistics, Poisson processes and

multiple Wiener integrals. Annals of Statistics 11 739–745 2.) van Ess, G. (1986) On the weak limit of elementary symmetric polynomials Annals

of Probability 14 No. 2 667–695 3.) Halász, G. and Székely, G. J. (1976) On the elementary symmetric polynomials of

independent random variables Acta Math. Acad. Sci. Hungar. 28 397–400 4.) Móri, T. F. and Székely, G. J. (1982) Asymptotical behaviour of symmetric poly-

nomial statistics The Annals of Probability 10 No. 1 124–131 5.) Parthasarathy, K. R. Probability Measures on Metric Spaces. Academic Press, New

York and London, 1967 6.) Raugi, A. (1979) Théorème de la limite centrale pour un produit semi-direct d’un

groupe de Lie résoluble 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.