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\centerline{\bf Central limit theorems for martingales.}

\centerline{\it by P\'eter Major}
\centerline{The Mathematical Institute of the Hungarian Academy of
Sciences} 

\medskip\noindent
{\narrower {\narrower
{\it Summary:}\/ In this note I study the central limit 
theorem for martingales, more precisely a slightly more 
general result when triangular arrays of martingale difference 
sequences and not only martingales are taken. Moreover, I 
shall present a slight generalization of this result when we 
take triangular arrays of almost (but not necessarily exact) 
martingale difference sequences. I present the basic notions 
which are needed to understand these results at the beginning 
of this paper. My goal is to present the most general known 
results in this field, and also to explain the main ideas 
behind their proofs. I shall present a modified version of 
the proof of B.~M.~Brown's paper {\it Martingale Central Limit 
Theorems}\/ in the journal {\it The Annals of Mathematical 
Statistics}\/ (1971) volume~42 No.~1 59--66, and briefly 
discuss the proof of Aryeh Dvoretzky in the paper 
{\it Asymptotic normality for sums of dependent random 
variables} in the II. volume of the {\it Sixth Berkeley 
Symposium}\/ pp. 513--535. The two results are similar, but 
they are proved by essentially different methods. I make a 
short comparison between these methods. I also try to explain
that a most essential ingredient of both proofs is that to get
a sharp version of the central limit theorem for triangular 
arrays of martingale difference sequences we have to work 
not with the variances of the terms in these arrays but with 
their conditional variances with respect to the past. This 
can be interpreted so that the conditional variances produce 
an `inner time' of the model which provides the natural 
time scaling in the investigation. I shall briefly discuss 
the functional central limit theorem version of the central 
limit theorem type result investigated in this paper, but I 
shall not work out all details of the proof. At this point
`the inner time' of the model appears again. It appears not
only in the proof but even in the formulation of the result.
At the end of this work I discuss L\'evy's characterization 
of Wiener processes by means of martingale type properties. 
This is a result closely related to the central limit 
theorem for martingales. \par} \par}

\beginsection 1. Introduction. Formulation of the main results.

In this note I discuss the generalization of the central limit theorem 
for normalized sums of independent random variables to the case when
we consider martingales instead of sums of independent random variables. 
To understand this result better first I recall the most general form 
of the central limit theorem for triangular arrays of independent 
random variables.

\medskip\noindent
{\bf Central limit theorem for triangular arrays of independent
random variables.} {\it Let a triangular array $X_{k,j}$, of 
independent random variables, i.e. a set of random variables 
$X_{k,j}$, $k=1,2,\dots$, $1\le j\le k_n$, indexed by a pair of 
positive integers be given which satisfies the following property~(a).

\medskip
\item{(a)} The random variables in the $k$-th row, i.e. the 
random variables $X_{k,1},\dots,X_{k,n_k}$ are independent of each 
other for all $k=1,2,\dots$, and also the identity $EX_{k,j}=0$ 
holds for all indices $k=1,2,\dots$ and $1\le j\le n_k$.

\medskip
Let this triangular array satisfy also the conditions

\medskip
\item{(b)} The sum of the variances of the random variables in
the $k$-th row of the triangular array tends to~1 as $k\to\infty$, 
i.e.,
$$
\limm_{k\to\infty}\summ_{j=1}^{n_k}EX_{k,j}^2=1
$$
\item{(c)}
The triangular array satisfies the so-called Lindeberg condition, i.e.
$$
\limm_{k\to\infty}\summ_{j=1}^{n_k}EX_{k,j}^2I(|X_{k,j}|>\e)=0\quad
\text{for all numbers } \e>0. \tag1.1
$$
(Here, and also in the subsequent formulas $I(A)$ denotes the indicator
function of a set $A$.)

\medskip
Then the random sums $S_k=\summ_{j=1}^{n_k}X_{k,j}$ converge in 
distribution to the standard normal distribution as $k\to\infty$.}

\medskip
In this note I discuss a central theorem for triangular series 
of not necessarily independent random variables which satisfy 
similar but weaker conditions than the conditions imposed in
the above result. We do not demand that the random variables in a
row of the triangular array should be independent, we only assume
that they constitute a martingale difference sequence. This is a
weakened version of condition (a) of the previous theorem.
We replace condition~(b) of this result by the assumption that
the sum of the conditional variances of the elements of the random 
variables in the $k$-th row of the triangular array with respect
to the past tend to 1 as $k\to\infty$. Finally, we need a weakened
version of the Lindeberg condition formulated in condition~(c).
The precise result is formulated in the following theorem.

\medskip\noindent
{\bf Central limit theorem for triangular arrays of martingale
difference sequ\-en\-ces.} {\it Let a sequence of random variables 
$X_{k,1},\dots,X_{k,n_k}$ be given for all integers $k=1,2,\dots$ 
together with an increasing sequence of $\sigma$-algebras 
$\Cal F_{k,0}\subset\Cal F_{k,1}\subset\cdots\subset\Cal F_{k,n_k}
\subset\Cal A$ in a probability space $(\Omega,\Cal A,P)$ which 
satisfies the following conditions:

\medskip
\item{(a)} For each number $k=1,2,\dots$, the random 
variables $X_{k,j}$, $1\le j\le n_k$, together with the 
$\sigma$-algebras $\Cal F_{k,j}$, $0\le j\le n_k$, constitute
a martingale difference sequence, i.e. the random variable 
$X_{k,j}$ is measurable with respect to the $\sigma$-algebra
$\Cal F_{k,j}$, and $E(X_{k,j}|\Cal F_{k,j-1})=0$ with 
probability~1 for all indices $1\le j\le n_k$.
\item{(b)} $EX_{k,j}^2<\infty$ for all indices $k=1,2,\dots$
and $1\le j\le n_k$, and the conditional variances defined as
$\sigma_{k,j}^2=E(X_{k,j}^2|\Cal F_{k,j-1})$, $k=1,2,\dots$,
$1\le j\le n_k$, satisfy the relation
$$
\sum_{j=1}^{n_k}\sigma^2_{k,j}\Rightarrow1 \quad\text{if }
k\to\infty. \tag1.2
$$
(Here and in the subsequent part of the paper $\Rightarrow$ 
denotes stochastic convergence.)
\item{(c)} The following Lindeberg type condition holds:
$$
\sum_{j=1}^{n_k}E(X^2_{k,j}I(|X_{k,j}|>\e)|\Cal F_{k,j-1})\Rightarrow0,
\quad\text{if } k\to\infty \tag1.3
$$
for all numbers $\e>0$.

\medskip
Under these conditions the random sums
$S_k=\summ_{j=1}^{n_k}X_{k,j}$, $k=1,2,\dots$, tend in distribution
to the standard normal distribution as $k\to\infty$.}

\medskip
The following generalization of this result where we consider
triangular arrays of `almost martingale difference sequences' also
holds.

\medskip\noindent
{\bf Central limit theorem for triangular arrays of almost 
martingale difference sequences.} {\it For each number $k=1,2,\dots$ 
let a sequence of random variables $X_{k,1}$,\dots, $X_{k,n_k}$ 
be given together with a sequence of increasing $\sigma$-algebras
$\Cal F_{k,0}\subset\Cal F_{k,1}\subset\cdots\subset \Cal F_{k,n_k}$,
such that the random variable $X_{k,j}$ is measurable with respect
to the $\sigma$-algebra $\Cal F_{k,j}$ for all indices $k=1,2,\dots$ 
and $1\le j\le n_k$, and the conditional expectations
$\mu_{k,j}=E(X_{k,j}|\Cal F_{k,j-1})$ are small in the following 
sense:
$$
\sum_{j=1}^{n_k}\mu_{k,j}\Rightarrow0 \quad\text{if }
k\to\infty. \tag1.4
$$
If the random variables $X_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
and $\sigma$-algebras $\Cal F_{k,j}$, $k=1,2,\dots$, $0\le j\le n_k$, 
satisfy conditions (1.2), (1.3) and (1.4) with the modification 
that in the present case we define the conditional variance 
$\sigma^2_{k,j}$ in formula~(1.2) as
$$
\sigma^2_{k,j}=E\((X_{k,j}-\mu_{k,j})^2|\Cal F_{k,j-1}\)=
E(X_{k,j}^2|\Cal F_{k,j-1})-\mu_{k,j}^2, \tag1.5
$$
then the random sums $S_k=\summ_{j=1}^{n_k}X_{k,j}$, 
$k=1,2,\dots$, converge in distribution to the standard normal 
distribution as $k\to\infty$.}

\medskip
The above results are the strongest known versions of the
central limit theorem for martingale difference or almost
martingale difference sequences. It may be worth mentioning
that in condition~(1.2) we only demanded that the sum of the
conditional second moments $\sigma^2_{k,j}$ (summing up for
the indices~$j$ with a fixed number~$k$) should tend to~1 as 
$k\to\infty$, but we did not impose such a condition which 
would imply that the conditional second moments
$\sigma^2_{k,j}=E(X_{k,j}^2|\Cal F_{k,j-1})$ are close to
the second moments $d_{k,j}^2=EX_{k,j}^2$. The proof of the
results under such relatively weak conditions demands finer
arguments. I know of two papers where the central limit theorem
was proved under such conditions. One of them is the work of
B.~M.~Brown {\it Martingale Central Limit Theorems}\/ in the
journal {\it The Annals of Mathematical Statistics}\/ (1971) 
volume~42 No.~1 59--66. The other one is the work of Aryeh 
Dvoretzky {\it Asymptotic normality for sums of dependent 
random variables} in the II. volume of the {\it Sixth Berkeley 
Symposium}\/ at pages 513--535. In these two works the 
difficulties arising during the proof are overcome by means 
of different methods. Brown's method is simpler, and it seems 
more appropriate in the investigation of more general limit 
theorem problems. Hence I describe here a slightly modified 
version of this proof. Although Brown's paper deals only with 
the central limit theorem for martingales, i.e. it does not 
investigate limit theorem for triangular arrays of martingale 
difference sequences, the application of his method in this 
more general case causes no problem. In the third section of 
this note I briefly compare Brown's and Dvoretzky's methods. 
I shall also discuss a result that can be considered as the 
functional central limit theorem version of the central limit 
theorem for triangular arrays of martingale difference 
sequences. Brown's paper contains a similar result when only 
appropriately normalized martingales are considered. The 
functional central limit theorem for triangular arrays of 
martingale difference sequences deserves special attention, 
because in the formulation of this result such new phenomena 
have to be taken into consideration which could be disregarded 
in the problem studied by Brown. I shall formulate this result, 
but omit the proof. I do not discuss the technical problems, 
I shall only explain some important ideas of the proof.

I shall finish this work with an Appendix where I present a
result of Paul L\'evy's characterization of Wiener processes.
This is an important result which is closely related to the
central limit theorem for martingales.

At the end of this introduction I make a short remark about the
Lindeberg type condition~(1.3) of the central limit theorem for
triangular arrays of martingale difference sequences. 
Formula~(1.3) follows from the Lindeberg condition presented 
in formula~(1.1), because
$$
\sum_{j=1}^{n_k}EX^2_{k,j}I(|X_{k,j}|>\e)
=E\(\sum_{j=1}^{n_k}E(X^2_{k,j}I(|X_{k,j}|>\e)|\Cal F_{k,j-1})\).
$$
Hence formula~(1.1) implies that the left-hand side expression
in~(1.3) tends to zero even in $L_1$-norm. The statement in 
the opposite direction does not hold. Such triangular arrays 
of martingale difference sequences can be constructed which 
satisfy relation~(1.3) but do not satisfy relation~(1.1). On 
the other hand, in the first step of the proof of the central 
limit theorem for martingale difference sequences we reduce 
the proof to such a special case where we may assume that the 
triangular array of martingale difference sequences has some 
additional nice properties. In particular condition~(1.3) 
implies condition~(1.1) for such triangular arrays.

\beginsection 2. The proof of the results.

{\it Proof of the central limit theorem for triangular arrays
of martingale difference sequences.}
First I show that the proof of this theorem can be reduced to
the special case when the elements of the triangular arrays 
satisfy beside the conditions of the theorem also the relation
$$
\sum_{j=1}^{n_k} \sigma^2_{k,j}\le 2 \quad \text{with 
probability 1 for all } k=1,2,\dots. \tag2.1
$$
(Actually we could write in the inequality of formula~(2.1) an 
arbitrary constant $C>1$ instead of~2.)

To prove this statement let us introduce for each $k=1,2,\dots$ 
the stopping time
$$
\tau_k=\min\(n_k,\;\max\left\{j\colon\; \summ_{l=1}^j\sigma_{k,l}^2
\le2\right\}\)\quad (\tau_k=0 \text{ if } \sigma^2_{k,1}>2.) \tag2.2
$$
and the random variables
$$
\bar X_{k,j}=\left\{
\aligned
&X_{k,j}, \quad \text{if } j\le \tau_k, \\ 
&\;0, \qquad  \text{if } j>\tau_k,
\endaligned \right. \qquad 1\le j\le n_k,
$$
The above defined random variable $\tau_k$ is really a 
stopping time with respect to the system of $\sigma$-algebras 
$\Cal F_{k,j}$, $0\le j\le n_k$, since the random variable 
$\sigma^2_{k,j+1}$ is $\Cal F_{k,j}$ measurable. Thus we can 
decide at time $j$ which of the events $\{\tau_k\le j\}$ or 
$\{\tau_k\ge j+1\}$ really occurred. (Let us observe that 
$\sigma^2_{k,j+1}$ is an $\Cal F_{k,j}$ and not only an 
$\Cal F_{k,j+1}$ measurable random variable.) Let us 
also introduce the random variables 
$\bar\sigma_{k,j}^2=E(\bar X_{k,j}^2|\Cal F_{k,j-1})$, 
$k=1,2,\dots$, $1\le j\le n_k$. For a fixed
number $k$ the sequence of random variables $\bar X_{k,j}$, 
$1\le j\le n_k$, together with the $\sigma$-algebras
$\Cal F_{k,j}$,  $0\le j\le n_k$, constitute a martingale
difference sequence. Furthermore, 
$\bar\sigma^2_{k,j}(\oo)=\sigma^2_{k,j}(\oo)$ if
$\tau_k(\oo)\ge j$, and $\bar\sigma^2_{k,j}(\oo)=0$ if
$\tau_k(\oo)< j$. Indeed, $E(\bar X_{k,j}|\Cal F_{k,j-1})=
E(X_{k,j}I(\tau_k\ge j)|\Cal F_{k,j-1})= I(\tau_k\ge j)
E(X_{k,j}|\Cal F_{k,j-1})=0$, and $\bar\sigma^2_{j,k}=
E(X_{k,j}^2 I(\tau_k\ge j)|\Cal F_{k,j-1})= I(\tau_k\ge j)
E(X_{k,j}^2|\Cal F_{k,j-1})=I(\tau_k\ge j)\sigma^2_{k,j}$.

Besides, relation (1.3) remains valid if we replace the random
variables $X_{k,j}$ by the random variables $\bar X_{k,j}$ 
in it, since $|\bar X_{k,j}|\le |X_{k,j}|$. Further
$\limm_{k\to\infty}P(\tau_k=n_k)=1$ because of relation~(1.2).

The above considerations show that relation~(1.2) remains 
valid if we replace the random variables $\sigma^2_{k,j}$ 
by $\bar\sigma^2_{k,j}$, and also relation~(2.1) holds in 
this case. Finally, the random sums  
$\bar S_k=\summ_{j=1}^{n_k}\bar X_{k,j}$, $k=1,2,\dots$, 
satisfy the relation $\bar S_k-S_k\Rightarrow0$ if $k\to\infty$, 
since $\bar S_k=S_k$ if $\tau_k=n_k$. Because of these 
relations it is enough to prove the central limit theorem for 
triangular arrays of martingale difference sequences for the 
triangular array $\bar X_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, 
instead of the triangular array $X_{k,j}$, and the random 
variables $\bar\sigma^2_{k,j}$ take the role of the random 
variables $\sigma^2_{k,j}$ in the formulation of the 
condition of the theorem in this case. We shall work with 
these random variables in the subsequent part of the paper, 
only we shall omit the sign bar in our notation. In such a way 
we are working with a triangular array of martingale difference 
sequences which satisfies the conditions of the theorem we want 
to prove together with relation~(2.1).

Relations (1.3) and (1.2) imply the Lindeberg condition~(1.1) for
the triangular array $X_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$,
since by Lebesgue's dominated convergence theorem  and the inequality
$$
\summ_{j=1}^{n_k}E(X^2_{k,j}I(|X_{k,j}|>\e)|\Cal F_{k,j-1})
\le\summ_{j=1}^{n_k}\sigma_{k,j}^2\le2
$$ 
also the relation
$$
\sum_{j=1}^{n_k}EX^2_{k,j}I(|X_{k,j}|>\e)=
E\(\sum_{j=1}^{n_k}E(X^2_{k,j}I(|X_{k,j}|>\e)|\Cal F_{k,j-1})\)\to0
\quad \text{if } k\to\infty 
$$
holds for all $\e>0$. It can be shown similarly that if 
relation~(2.1) holds, then we can write in~(1.2) $L_1$-convergence 
instead of stochastic convergence, and
$$
\limm_{k\to\infty}\summ_{j=1}^{n_k}E\sigma_{k,j}^2=
\limm_{k\to\infty}\summ_{j=1}^{n_k}EX_{k,j}^2=1. \tag2.3
$$

The central limit theorem we want to prove is equivalent to 
the relation
$$
\lim_{k\to\infty}Ee^{itS_k}=e^{-t^2/2}\quad \text{for all real
numbers } t. \tag2.4
$$
We show with the help of formula~(2.1) that relation (2.4) 
follows from the statement 
$$
\lim_{k\to\infty}E e^{itS_k+t^2U_k/2}=1\quad
\text{for all real numbers } t, \tag2.5
$$
where $U_k=\summ_{j=1}^{n_k}\sigma^2_{k,j}$, $k=1,2,\dots$. 
(A direct proof of formula~(2.5) is simpler.)

Indeed, by formula (1.2) $U_k\Rightarrow1$ if $k\to\infty$, and 
$0\le U_k\le2$ for all numbers $k=1,2,\dots$ because of 
formula~(2.1). Hence $e^{itS_k+t^2U_k/2}-e^{itS_k+t^2/2}\Rightarrow0$ 
for all real numbers $t$ if $k\to\infty$, and
$|e^{itS_k+t^2U_k/2}- e^{itS_k+t^2/2}|\le2\cdot2^{1+t^2}$. Hence by
Lebesgue's dominated convergence theorem
$\limm_{k\to\infty}E(e^{itS_k+t^2U_k/2}- e^{itS_k+t^2/2})=0$. Formula~(2.4)
follows from this statement and relation~(2.5).

To prove relation~(2.5) first we show that there exists a number
$K(t)>0$ depending only on the parameter~$t$ for which the inequality
$$
|E e^{itS_k+t^2U_k/2}-1|\le K(t)\summ_{j=1}^{n_k}
E\left|e^{t^2\sigma_{k,j}^2/2}E\(e^{itX_{k,j}}|\Cal F_{k,j-1}\)-1
\right|. \tag2.6
$$
holds. Indeed, let us introduce the random variables
$$
S_{k,j}=\sum_{l=1}^j X_{k,l},\quad
U_{k,j}=\sum_{l=1}^j \sigma^2_{k,l}, \quad 1\le j\le n_k
$$
and $S_{k,0}=0$, $U_{k,0}=0$ for all indices $k=1,2,\dots$. Then
we have $S_{k,n_k}=S_k$, $U_{k,n_k}=U_k$, and
$$
\align
Ee^{itS_k+t^2U_k/2}-1&= \sum_{j=1}^{n_k}
E\(e^{itS_{k,j}+t^2U_{k,j}/2}
-e^{itS_{k,j-1}+t^2U_{k,j-1}/2}\)\\
&= \sum_{j=1}^{n_k}
Ee^{itS_{k,j-1}+t^2U_{k,j-1}/2}
E\left.\(e^{itX_{k,j}+t^2\sigma_{k,j}^2/2}-1\right|\Cal F_{k,j-1}\).
\endalign
$$
Since the random variable $e^{itS_{k,j-1}+t^2U_{k,j-1}/2}$ 
is bounded, it is smaller than some number $K(t)$ depending 
only on the parameter $t$, it follows from the above identity that
$$
|Ee^{itS_k+t^2U_k/2}-1|\le K(t) \sum_{j=1}^{n_k}
E\left|E\(e^{itX_{k,j}+t^2\sigma_{k,j}^2/2}-1|\Cal F_{k,j-1}\)\right|,
$$
and as
$E\(e^{itX_{k,j}+t^2\sigma_{k,j}^2/2}-1|\Cal F_{k,j-1}\)=
e^{t^2\sigma_{k,j}^2/2}E(e^{itX_{k,j}}|\Cal F_{k,j-1})-1$, this
implies the estimate~(2.6).

To prove formula~(2.5) with the help of inequality~(2.6) we have
to give a good estimate on the expressions
$E\left|e^{t^2\sigma_{k,j}^2/2}E\(e^{itX_{k,j}}|\Cal F_{k,j-1}\)-1
\right|$. The following heuristic argument is behind the estimation 
we shall apply in the study of these expressions. The Taylor 
expansion of the function $e^{t^2\sigma_{k,j}^2/2}$ is of the form 
$1+\frac{t^2\sigma^2_{k,j}}2+\cdots$, while the Taylor expansion
of the function $E(e^{itX_{k,j}}|\Cal F_{k,j-1})$ (because of the 
relation $E(X_{k,j}|\Cal F_{k,j-1})=0$) is of the form
$$
E(e^{itX_{k,j}}|\Cal F_{k,j-1})=1+E(itX_{k,j}|\Cal F_{k,j-1})
-\frac{E(t^2X_{k,j}^2|\Cal F_{k,j-1})}2+\cdots
=1-\frac{t^2\sigma^2_{k,j}}2+\cdots.
$$
Hence the constant, the first and second terms in the Taylor
expansion of the function
$e^{t^2\sigma_{k,j}^2/2}E\(e^{itX_{k,j}}|\Cal F_{k,j-1}\)-1$
disappears, which indicates that this function is small. We
expect that because of this facts we can give a good bound
on the right-hand side of formula~(2.6). In the estimation of
this expression we shall exploit that the random variables
$\sigma^2_{k,j}$ are small because of formulas~(2.1) and~(1.1).

The expression $e^{t^2\sigma_{k,j}^2/2}$  can be written in the
form $e^{t^2\sigma_{k,j}^2/2}=
1+\frac{t^2\sigma^2_{k,j}}2+\eta_{k,j}^{(1)}$ with an appropriate
random variable $\eta_{k,j}^{(1)}$ which satisfies the inequality
$|\eta_{k,j}^{(1)}|\le  K_1(t)\sigma_{k,j}^4$ with some number
$K_1(t)$ depending only on the parameter~$t$, because 
$\sigma^2_{k,j}\le2$ by formula~(2.1). We can estimate the expression
$$
\eta_{k,j}^{(2)}=E\left.\(e^{itX_{k,j}}-1+\frac{t^2X_{k,j}^2}2
\right|\Cal F_{k,j-1}\)
$$
in a similar way. To do this let us fix a small number $\e>0$, 
and show that the inequality
$$
\left|e^{itX_{k,j}}-1-itX_{k,j}+\frac{t^2X_{k,j}^2}2\right|\le
\alpha(X_{k,j})=\alpha_{\e,t}(X_{k,j})
$$
holds with $\alpha(x)=t^2x^2I(|x|>\e)+\frac\e6|t|^3x^2I(|x|\le\e)$. 
Indeed, we get this estimate by bounding the expression 
$\left|e^{itx}-1-itx+\frac{t^2x^2}2\right|$ by $t^2x^2$ if $|x|>\e$ 
and by $\frac{|t|^3|x|^3}6\le\e\frac{|t|^3x^2}6$ if $|x|\le\e$. 
By exploiting the relation $E(X_{k,j}|\Cal F_{k,j-1})=0$ and 
taking the conditional expectation of the random variables in the 
last inequality with respect to the $\sigma$-algebra 
$\Cal F_{k,j-1}$ we get the following inequality:
$$ 
\align
|\eta_{k,j}^{(2)}|&=
\left|E\left.\(e^{itX_{k,j}}-1-itX_{k,j}+\frac{t^2X_{k,j}^2}2
\right|\Cal F_{k,j-1}\)\right| \\
& \le
E\left.\(\left|e^{itX_{k,j}}-1-itX_{k,j}+\frac{t^2X_{k,j}^2}2
\right|\right|\Cal F_{k,j-1}\)\\
&\le E(\alpha(X_{k,j})|\Cal F_{k,j-1}) 
\le t^2E(X_{k,j}^2I(|X_{k,j}|>\e)|\Cal F_{k,j-1})
+\frac\e6 |t|^3\sigma^2_{k,j}.
\endalign
$$

Since $\sigma^2_{k,j}\le2$ by formula (2.1), both
$\eta^{(1)}_{k,j}$ and $\eta^{(2)}_{k,j}$ are bounded
random variables (with a bound depending only on the 
parameter $t$), and the above estimates imply that
$$
\align
&\left|e^{t^2\sigma_{k,j}^2/2}E\(e^{itX_{k,j}}|\Cal F_{k,j-1}\)-1\right|
=\left|\(1+\frac{t^2\sigma_{k,j}^2}2+\eta_{k,j}^{(1)}\)
\(1-\frac{t^2\sigma_{k,j}^2}2+\eta_{k,j}^{(2)}\)-1\right| \\
&\qquad\le t^4\sigma^4_{k,j}+K_3(t)\(|\eta^{(1)}_{k,j}|
+|\eta_{k,j}^{(2)}|\)\\
&\qquad \le K_4(t)(\sigma^4_{k,j}+
E(X_{k,j}^2I(|X_{k,j}|>\e)|\Cal F_{k,j-1})+\e \sigma^2_{k,j}).
\endalign
$$
Let us take the expectation of the left-hand side and right-hand side
expression in the last inequality and sum up for all indices
$1\le j\le n_k$. The inequality obtained in such a way 
together with formula~(2.6) imply that
$$
|E e^{itS_k+t^2U_k/2}-1|\le
 K_5(t)\(\sum_{j=1}^{n_k}E\sigma^4_{k,j}+
\sum_{j=1}^{n_k}EX_{k,j}^2I(|X_{k,j}|>\e)
+\e\sum_{j=1}^{n_k} E\sigma^2_{k,j}\). \tag2.7
$$
To estimate the first sum at the right-hand side of~(2.7)
let us make the following estimate:
$$
\align
E\sigma^4_{k,j}&=E\((EX^2_{k,j}I(|X_{k,j}|>\e)|\Cal F_{k,j-1})
+(EX_{k,j}^2I(|X_{k,j}|\le\e)|\Cal F_{k,j-1})\)^2\\
&\le2\(E(EX_{k,j}^2I(|X_{k,j}|>\e)|\Cal F_{k,j-1})^2
+E(EX_{k,j}^2I(|X_{k,j}|\le\e)|\Cal F_{k,j-1})^2\)\\
&\le2E\sigma_{k,j}^2E(X_{k,j}^2I(|X_{k,j}|>\e)|\Cal F_{k,j-1})
+2\e^2 E(EX_{k,j}^2I(|X_{k,j}|\le\e)|\Cal F_{k,j-1}) \\
&\le4EX_{k,j}^2I(|X_{k,j}|>\e)+2\e^2 E\sigma_{k,j}^2.
\endalign
$$
(Let us choose in this estimate the same number $\e>0$ as in 
formula~(2.7).) With the help of this estimate we can formulate
the following consequence of relation~(2.7).
$$
|E e^{itS_k+t^2U_k/2}-1|\le
 K_6(t)\(\sum_{j=1}^{n_k}EX_{k,j}^2I(|X_{k,j}|>\e)
+\e\sum_{j=1}^{n_k} E\sigma^2_{k,j}\). \tag2.8
$$
As formula~(2.8) holds for all numbers $\e>0$, hence relations
(1.1), (2.3) an (2.8) imply formula~(2.5). Thus we have proved the
central limit theorem for triangular arrays of martingale difference
sequences. 

\medskip
I turn to the proof of the central limit theorem for triangular
arrays of almost martingale difference sequences. A natural idea
would be to reduce the proof to the already proved the central limit
theorem for triangular arrays of martingale difference sequences 
by means of the introduction of the random variables 
$\bar X_{k,j}=X_{k,j}-\mu_{k,j}$. The main problem in the
application of such an argument would be the control of the
Lindeberg condition~(c) of this theorem. To overcome this
difficulty it is worth refining this argument, and to combine
it with an appropriate version of the truncation argument applied
at the beginning of the previous proof.

\medskip\noindent
{\it Proof of the central limit theorem for triangular arrays of 
almost martingale difference sequences.} Let us introduce a 
slightly modified version of the stopping times introduced in
formula~(2.2). In this modified definition of the stopping time
we work with the random variables $\sigma^2_{k,j}$ defined in
formula~(1.5). Let us also introduce the random variables
$\bar X_{k,j}=X_{k,j}I(\tau_k\ge j)$,
$\bar\mu_{k,j}=E(\bar X_{k,j}|\Cal F_{k,j-1})$ and
$\bar\sigma_{k,j}^2=E(\bar X^2_{k,j}|\Cal F_{k,j-1})-\bar\mu^2_{k,j}$
$k=1,2,\dots$, $1\le j\le n_k$. Then 
$\limm_{k\to\infty}P(\tau_k=n_k)=1$, hence the probability of 
the events that $\bar X_{k,j}=X_{k,j}$, $\bar\mu_{k,j}=\mu_{k,j}$ and 
$\bar\sigma^2_{k,j}=\sigma^2_{k,j}$ for all indices $1\le j\le n_k$ 
tends to~1 as $k\to\infty$, and $|\bar X_{k,j}|\le |X_{k,j}|$ for 
all pairs of indices~$(j,k)$. Hence we can, similarly to the
argument in the proof of the previous result, reduce the proof
to that special case, when beside the conditions of the theorem also
relation~(2.1) holds.

Let us omit the sign bar from the notation of the random variables
$\bar X_{k,j}$, $\bar\sigma_{k,j}^2$ and $\bar\mu_{k,j}$ defined
below, and let us define with their help the random variables 
$\tilde X_{k,j}=X_{k,j}-\mu_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$.
We want to show that the triangular array of martingale difference
sequences consisting of the random variables $\tilde X_{k,j}$, 
and $\sigma$-algebras $\Cal F_{k,j}$, $k=1,2,\dots$, 
$0\le j\le n_k$, satisfies the conditions of the central limit
theorem for martingale difference series. This system clearly
satisfies conditions~(a) and~(b) of this theorem, but the 
validity of condition~(c), i.e. of the Lindeberg condition demands
some explanation. Since we know that the triangular array $X_{k,n}$
satisfies the original version of the Lindeberg condition formulated
in~(1.1), to complete the proof of our theorem it is enough to 
prove the following statement. If a triangular array 
$(X_{k,j},\Cal F_{k,j})$ satisfies relation~(1.1), and
$\mu_{k,j}=EX_{k,j}$, then

$$
\limm_{k\to\infty}\summ_{j=1}^{n_k}E(X_{k,j}-\mu_{k,j})^2
I(|X_{k,j}-\mu_{k,j}|>\e)=0\quad \text{for all numbers } \e>0.
$$

This relation is a direct consequence of the following lemma which
agrees with Lemma~3.3 of Dvoretzky's article mentioned in the
Introduction. (There is a small difference between the two results.
At the left-hand side of this estimate I wrote a multiplying constant~8
instead of the multiplying constant~4 in Dvoretzky's lemma, because
this made the proof simpler. But the value of this multiplying
constant has no importance in our investigation.)

\medskip\noindent
{\bf Lemma.} {\it Let $X$ be a random variable with finite second
moment, $\Cal F\subset \Cal A$ a $\sigma$-algebra and 
$\mu=E(X|\Cal F)$ in a probability space $(\Omega,\Cal A,P)$. Then
$$
8EX^2I(|X|>\e)\ge E(X-\mu)^2I(|X-\mu|>2\e) \quad
\text{for all numbers } \e>0. \tag2.9
$$
}

%\medskip
\noindent
{\it Remark:}\/ We know that $E(X-E(X|\Cal F))^2\le EX^2$
for all random variables~$X$ with finite second moment.
If we replace the random variable $X$ by a truncation of it, 
then this inequality may loose its validity. But an 
appropriate weakened version of it which may suffice for 
our goal remains valid. This can be considered as 
`the message' of this lemma.

\medskip\noindent
{\it The proof of the Lemma.}\/ By taking the conditional 
distribution of the random variable~$X$ with respect to the
$\sigma$-algebra $\Cal F$, and by denoting with $E$ the
expected value with respect to this (random) conditional
measure formula~(2.9) can be reduced to the following 
inequality:
$$
8EX^2I(|X|>\e)\ge E(X-EX)^2I(|X-EX|>2\e) \quad
\text{for all numbers } \e>0,
$$
or we can rewrite this in an equivalent form with the help of the
transformation $Y=X-EX$ as
$$
8E(Y+c)^2I(|Y+c|>\e)\ge EY^2I(|Y|>2\e) \tag2.10
$$
for all real numbers $c$ and $\e>0$ if $EY=0$ and $EY^2<\infty$.

Inequality~(2.10) can be reduced to a simpler statement. It is
enough to consider the special case when the distribution of $Y$
is of the form $P(Y=Aq)=p$ and $P(Y=-Ap)=q$ with some
numbers $A>0$ and $0\le p,q\le1$, $p+q=1$, since the distribution
of arbitrary random variable with expectation zero (and finite
second moment) can be well approximated by an appropriate linear 
 combination $w_1F_1+\cdots+w_nF_n$, $w_1+\cdots+w_n=1$,
$w_j>0$, $1\le j\le n$, of distribution functions $F_j$ of
random variables of this form. We may also assume with the
possible modification of the parameters that $A=1$, and 
$q\ge\frac12\ge p\ge0$.

In this special case inequality~(2.10) clearly holds if
$\e\ge\frac q2$, because the right-hand side of the inequality
equals zero in this case. If $0\le \e<\frac q2$, then the 
right-hand side of formula~(2.10) is less than or equal to
$EY^2=pq^2+p^2q=pq$, and it is enough to show that 
$8E(Y+c)^2I(|Y+c|>\e)\ge pq$ for all real numbers $c$ if
$0\le\e<\frac q2$.

If $c\ge-\frac q2$, and $0\le\e<\frac q2$, then
$q+c\ge\frac q2>\e$, and
$8E(Y+c)^2I(|Y+c|>\e)\ge 8P(Y=q)(q+c)^2=8p(q+c)^2\ge2pq^2\ge pq$.
If $c<-\frac q2$, and $0\le\e<\frac q2$, then
$|-p+c|>\frac q2\ge\e$, and
$8E(Y+c)^2I(|Y+c|>\e)\ge 8P(Y=-p)(p+|c|)^2=8q(p+|c|)^2\ge2q^3\ge pq$.
The Lemma is proved.

\beginsection 3. Some additional remarks. The functional central
limit theorem.

I start the comparison of the various proofs of the central limit 
theorem for martingales and similar objects with a short
discussion of the traditional proof of such results.

Let us consider a triangular array of martingale difference sequences
$X_{k,j}$, $k=1,2,\dots$, $1\le j\le n_k$, together with a set of
increasing (for a fixed number $k$) sequence of $\sigma$-algebras 
$\Cal F_{k,j}$, $k=1,2,\dots$, $0\le j\le n_k$, such that the sequence
of pairs $(X_{k,j},\Cal F_{k,j})$, $1\le j\le n_k$, constitute a
martingale difference sequence. Let us define the random sums  
$S_k=\summ_{j=1}^{n_k}X_{k,j}$, $k=1,2,\dots$. We want to show that
under appropriate conditions the sequence of random sums $S_k$, 
$k=1,2,\dots$, converge in distribution to the standard normal
distribution, or in an equivalent formulation
$\limm_{k\to\infty}Ee^{itS_k}=e^{-t^2/2}$ for all real number~$t$.

The classical, traditional proof is based on the following argument.
Let us introduce for all indices $1\le j\le n_k$ such independent 
random variables $Y_{k,1},\dots,Y_{k,n_k}$ which are independent 
also of the $\sigma$-algebra $\Cal F_{k,n_k}$, define the random 
sum $T_k=\summ_{j=1}^{n_k}d_{k,j}Y_j$ with $d^2_{k,j}=EX_{k,j}^2$, 
and let us show that $\limm_{k\to\infty}E(e^{itS_k}-e^{iT_k})=0$ 
under appropriate conditions. We try to prove this statement 
in such a way that first we replace in the sum $S_k$ the term 
$X_{k,n_k}$ by $d_{k,n_k}Y_{k,n_k}$, then the term $X_{k,n_k-1}$ 
by $d_{k,n_k-1}Y_{k,n_k-1}$, and we follow this procedure till
we get to the sum $T_k$. During this procedure we give a good
estimate about how much the characteristic function of the sum
changed during each replacement. In a more explicit formulation 
we apply the following procedure. Let us define the random sums
$S_{k,j}=\summ_{l=1}^j X_{k,l}+\summ_{l=j+1}^{n_k}d_{k,l} Y_{k,l}$, 
for $1\le j\le n_k-1$, and put $S_{k,0}=T_k$ and $S_{k,n_k}=S_k$. 
Then we have
$$
E(e^{itS_k}-e^{itT_k})=\sum_{j=1}^{n_k}
E(e^{itS_{k,j}}-e^{itS_{k,j-1}}). \tag3.1
$$
We try to prove the central limit theorem with the help of this
formula and a good estimate on the expressions 
$|E(e^{itS_{k,j}}-e^{iS_{k,j-1}})|$. It is not difficult to prove
that
$$
\aligned
\left|E(e^{itS_{k,j}}-e^{iS_{k,j-1}})\right|
&\le E\left|E(e^{itX_{k,j}}|\Cal F_{k,j-1})
-Ee^{itd_{k,j}Y_{k,j}}\right|\\
&=E\left|E(e^{itX_{k,j}}|\Cal F_{k,j-1})-e^{-t^2d_{k,j}^2/2}\right|.
\endaligned \tag3.2
$$
Let us observe that the system of formulas (3.1) and (3.2) is
similar to the inequality~(2.6). The expression at the right-hand
side of formula~(3.2) can be well estimated if we take the Taylor
series expansion of the function
$E(e^{itX_{k,j}}|\Cal F_{k,j-1})-e^{-t^2d^2_{k,j}/2}$
with respect to the variable~$t$. Let us observe the second order
term of this Taylor expansion equals
$\frac{t^2}2(d_{k,j}^2-\sigma^2_{k,j})$, where
$\sigma^2_{k,j}=E(X_{k,j}^2|\Cal F_{k,j-1})$. As we take the
expectation of the absolute value of the random variable at the
right-hand side of~(3.2) we can prove the central limit theorem
with the help of the above estimation the sum
$\summ_{j=1}^{n_k}|\sigma^2_{k,j}-d_{k,j}^2|$ is small for large
indices $k$. In such a way we can prove a result that is useful
in several cases. Nevertheless, it holds only under more 
restrictive conditions than the central limit theorem for 
triangular arrays of martingale differences formulated in 
the Introduction. We demanded there in the typical 
`non-degenerate' cases when 
$\limm_{k\to\infty}\summ_{j=1}^{n_k}d_{k,j}^2=1$ only the condition
$$
\sum_{j=1}^{n_k}\(\sigma^2_{k,j}-d_{k,j}^2\)\Rightarrow0
\quad\text{if } k\to\infty,
$$
which is an equivalent reformulation of formula~(1.2), i.e. we 
did not have to take the absolute value of the terms in the 
sum we considered.

The main merit of the proof of Brown and Dvoretzky is that the
authors of these proofs could prove the central limit theorem
for martingale difference sequences under weaker condition. 
To do this they had to work out a non-trivial refinement of the 
above sketched method.

Dvoretzky's proved the central limit theorem for triangular arrays
of martingale difference sequences similarly to the method explained
at the start of this section. But he replaced the terms 
$d_{k,l}Y_{k,l}$ by terms of the form $\sigma_{k,l}Y_{k,l}$ in the 
definition of the random sums $T_k$ and $S_{k,j}$. To prove the
central limit theorem after such a modification of the definition
of $T_k$ and $S_{k,j}$ we need such a version of formula~(3.2) 
where we replace the term $Ee^{itd_{k,j}Y_{k,j}}$ by 
$E(e^{it\sigma_{k,j}Y_{k,j}}|\Cal F_{k,j-1})=e^{-t^2\sigma^2_{k,j}/2}$ 
in the middle and right-hand side expressions of this relation.
Such an estimate enables us to prove the stronger version of the
central limit theorem formulated in the Introduction, since the
coefficient of the second order term in the corresponding Taylor
expansion equals zero.
 
Dvoretzky could prove the central limit theorem in such a way,
but only under the additional condition that 
$$
\sum_{j=1}^{n_k}\sigma_{k,j}^2=1 \quad \text{with probability 1
for all indices } k. \tag3.3
$$
He needed this condition to guarantee the independence of those
random variables and $\sigma$-algebras with which he worked
in the proof of the modified version of formula~(3.2). I omit
the precise formulation of this result. It is contained in
Lemma~3.2 of Dvoretzky's paper. (I would mention that if 
relation~(3.3) holds, then $T_k=\summ_{j=1}^{n_k}\sigma_{k,j}Y_j$ 
is a standard normal distributed random variable, since in this
case the conditional distribution of $T_k$ with respect to the
$\sigma$-algebra $\Cal F_{k,n_k}$ is the normal distribution
with expectation zero and second moment
$\summ_{j=1}^{n_k}\sigma_{k,j}^2=1$.)
 
After proving the central limit the theorem under the additional
condition~(3.3) Dvoretzky proved with the help of a stopping 
time similar to the stopping time defined in formula~(2.2) that 
the proof of the central limit theorem in the general case can 
be reduced to this special case.

The idea of Brown's proof, explained in this note, is very similar
to that of Dvoretzky. In this proof we estimate (with the notations
introduced in Section~2) the expression $Ee^{itS_k+t^2U_k/2}$ 
instead of the characteristic function $Ee^{itS_k}$. To do this
we have to guarantee that the expected value of the random
variables we are working with is finite, hence in the first step
of the proof we apply such a modification of the original 
triangular array of martingale difference sequences that makes this
possible. With the introduction of the stopping times $\tau_k$
in formula~(2.2) we apply some sort of truncation which enables
us to carry out our calculation. We know that the limit 
distribution of the original and modified random sums agree. 
Besides, the expected value $Ee^{itS_k+t^2U_k/2}$ defined with 
the help of the modified random variables $S_k$ and $U_k$ is 
finite, and even the random variables $\sigma^2_{k,j}$ and their 
partial sums are finite.

I remark that the modified triangular array of martingale difference
sequences defined with the help of the stopping time~(2.2) also 
satisfies the original version of the Lindeberg condition~(1.1),
hence the triangular array defined by the formula
$\tilde X_{k,j}=X_{k,j}I(|X_{k,j}|\le\e)-
E(X_{k,j}I(|X_{k,j}|\le\e)|\Cal F_{k,j-1})$, 
$k=1,2,\dots$, $1\le j\le n_k$, is such a small perturbation
of the original triangular array that satisfies the conditions
of the central limit theorem for triangular arrays of martingale
difference series, and it contains bounded random variables.
(Let us observe that the sums of the random variables 
$X_{k,j}-\tilde X_{k,j}=
X_{k,j}I(|X_{k,j}|>\e)-E(X_{k,j}I(|X_{k,j}|>\e)|\Cal F_{k,j-1})$
can be well bounded. Furthermore the Lindeberg condition~(1.1)
implies that the replacement of the random variables $X_{k,j}$ 
by the random variables $\tilde X_{k,j}$ causes a negligible 
small error. This observation has no importance in the proof
of the central limit theorem, but it may be useful in the proof
of the functional central limit theorem, since it enables us to
work with exponential moments.

We can understand the similarity of the method of Dvoretzky and
Brown with the help of the following observation. We can write,
applying the notation introduced at the beginning of this section, 
that 
$$
E\(e^{it\sigma_{k,j}Y_{k,j}+t^2\sigma^2_{k,j}/2}|\Cal F_{k,j-1}\)=
\left.E(e^{ituY_{k,j}+t^2u^2/2})\right|_{u=\sigma^2_{k,j}}=1
\tag3.4
$$
for all indices $k=1,2,\dots$, $1\le j\le n_k$. Let us introduce
the following counterparts $\bar S_{k,j}$, $\bar S_k$ and $\bar T_k$
of the previously defined random variables $S_{k,j}$, $S_k$ and $T_k$.
Put
$\bar S_{k,j}=\bar S_{k,j}(t)=\summ_{l=1}^j (it X_{k,l}
+\frac{t^2}2\sigma^2_{k,l})+
\summ_{l=j+1}^{n_k}(it\sigma_{k,l}Y_{k,l}+\frac{t^2}2\sigma^2_{k,l})$,
$1\le j\le n_k-1$, $\bar S_{k,0}=\bar S_{k,0}(t)=\bar T_k=
\summ_{l=1}^{n_k}(it\sigma_{k,j}Y_{k,j}+\frac{t^2}2\sigma^2_{k,j})$,
and
$\bar S_{k,n_k}=\bar S_{k,n_k}(t)=\bar S_k=\summ_{l=1}^{n_k} (it X_{k,j}
+\frac{t^2}2\sigma^2_{k,j})$. If we want to estimate  
the expression $Ee^{itS_k+t^2 U_k/2}=e^{\bar S_k}$ instead of
$Ee^{itS_k}$, then it is useful to write up the identity
$$
Ee^{\bar S_k}-1=E\(e^{\bar S_k}-e^{\bar T_k}\)=\sum_{j=1}^{n_k}
E\(e^{\bar S_{k,j}}-e^{\bar S_{k,j-1}}\)
$$
and to estimate its right-hand side. Let us observe that because of
relation~(3.4) in the formula~(2.5) of the previous section and in
the subsequent calculations we carried out such a program.

We can understand better the picture behind the central limit 
theorem for martingale difference sequences and Brown's proof 
for this theorem if we also discuss the functional central limit 
theorem version of this result. In an informal way the functional 
central limit theorem states that the random broken lines 
constructed with the help of the partial sums of the random 
variables from the $k$-th row of a triangular array of martingale 
difference sequences behave similarly to a Wiener process for 
large indices~$k$. The following statement expresses an important 
property of the Wiener process. If $W(u)$, $u\ge0$, is a Wiener 
process, then the stochastic process $Z_t(u)=e^{tW(u)-t^2u/2}$,
$u\ge0$, is a martingale. Hence $Ee^{Z_t(\tau)}=1$ for all nice
stopping time $\tau$. This statement hold not only for real but
also for complex numbers~$t$. It is natural to expect that if
we replace the Wiener process $W(u)$ with a stochastic process
$V(u)$ which is similar to it (in an appropriate sense), then
the relation $Ee^{tV(\tau)-t^2\tau}\sim1$ holds. In the proof
of the central limit theorem, in particular in the formulation
of formula~(2.5) we applied such an argument with a purely 
imaginary number~$t$.

To formulate the functional central limit theorem for triangular
arrays of martingale difference sequences I introduce some
notations.

For all numbers $k=1,2,\dots$ let us consider such a sequence of
random variables $X_{k,1},\dots,X_{k,n_k}$ together with a
sequence of increasing $\sigma$-algebras
$\Cal F_{k,0}\subset\Cal F_{k,1}\subset\cdots\subset\Cal F_{k,n_k}$
which satisfy the conditions of the central limit theorem for
triangular arrays of martingale difference sequences. Let us
define the partial sums
$$
S_{k,j}=\sum_{l=1}^j X_{k,j}\quad 1\le j\le n_k, \quad S_{k,0}=0,
\tag3.5
$$
the variances $d_{k,j}^2=EX_{k,j}^2$ and conditional variances 
$\sigma^2_{k,j}=E(X_{k,j}^2|\Cal F_{k,j-1})$, $1\le j\le n_k$.
Let us introduce with their help the following deterministic
set of points $z_{k,j}$ and random set of points $\zeta_{k,j}$
$$
z_{k,0}=0,\; z_{k,j}=\sum_{l=1}^j d^2_{k,j}, \qquad \zeta_{k,0}=0,\;
\zeta_{k,j}=\sum_{l=1}^j \sigma^2_{k,j}, \quad 1\le j\le n_k, \tag3.6
$$
on the positive half-line. With the help of these points we shall
define the following random broken line process $T_k(t)$ on the
interval $[0,z_{0,k}]$ and random broken line process $V_k(t)$ 
on the interval $[0,\zeta_{0,k}]$:
$$
\aligned
T(z_{k,j})=S_{k,j},
\quad \text{and } T_k(t)&=\frac{z_{k,j+1}-t}{z_{k,j+1}-z_{k,j}}S_{k,j}
+\frac{t-z_{k,j}}{z_{k,j+1}-z_{k,j}}S_{k,j+1}, \\
&\qquad\text{ if }
z_{k,j}\le t\le z_{k,j+1}, \quad 0\le j< n_k,
\endaligned \tag3.7
$$
and
$$
\aligned
V(\zeta_{k,j})=S_{k,j}, \quad \text{and } V_k(t)
&=\frac{\zeta_{k,j+1}-t}{\zeta_{k,j+1}-\zeta_{k,j}}S_{k,j}
+\frac{t-\zeta_{k,j}}{\zeta_{k,j+1}-\zeta_{k,j}}S_{k,j+1}, \\
&\qquad\text{ if }
\zeta_{k,j}\le t\le \zeta_{k,j+1}, \quad 0\le j< n_k.
\endaligned\tag3.8
$$
We also define the following rescaled versions of the
random broken line processes $T_k(\cdot)$ $V_k(\cdot)$
in the interval $[0,1]$.
$$
\tilde T_k(t)=T_k(tz_{k,n_k}), \qquad
\tilde V_k(t)=V_k(t\zeta_{k,n_k}), \quad 0\le t\le1 \tag3.9
$$
Next I formulate the functional central limit theorem for
triangular arrays of martingale difference sequences.

\medskip\noindent
{\bf Functional central limit theorem for triangular arrays of
martingale difference sequences.} {\it For all numbers 
$k=1,2,\dots$ let a sequence of random variables 
$X_{k,1},\dots,X_{k,n_k}$ be given together with a sequence of
increasing $\sigma$-algebras
$\Cal F_{k,0}\subset\Cal F_{k,1}\subset\cdots\subset\Cal F_{k,n_k}$
that satisfies the conditions of the central limit theorem for
martingale difference sequences. Let us consider the sequence
of broken line processes $\tilde V_k(t)$, $k=1,2,\dots$,  
defined in formulas (3.5), (3.6), (3.8) and (3.9). The random
broken line processes $\tilde V_k(t)$, $k=1,2,\dots$, can
be considered as random variables taking their values in the 
Banach space $C([0,1])$ of continuous functions in the interval
$[0,1]$, and this random variables converge weakly to the 
Wiener measure in the space $C([0,1])$ as $k\to\infty$.}

\medskip
I omit the detailed proof the central limit theorem, I only make
some remarks about how this can be done with the help of some
classical results. First we have to show that the finite
dimensional distributions of the random processes 
$\tilde V_k(t)$ converge. This can be done with the help of 
the central limit theorem for martingale difference
sequences by introducing some appropriate stopping times. 
We also have to prove a statement called the tightness property
in the literature. In the present case this means the proof of
some maximum type inequality. We can prove such inequalities
by exploiting that with the help of some truncation procedure
we can reduce the problem to that special case when the random
variables $X_{k,j}$ are bounded, moreover we may assume that
this bound is very small. The sequences 
$(S_{k,j},\Cal F_{k,j})$, $1\le j\le n_k$, are martingales.
This implies that the sequences $(e^{tS_{k,j}},\Cal F_{k,j})$ 
are submartingales, and we can apply for them the classical 
inequalities valid for submartingales. Besides, we can 
estimate the exponential moments $Ee^{t(S_{k,j'}-S_{k,j})}$, 
$1\le j\le j'\le n_k$, by means of the methods applied in 
Section~2, and in such a way we can prove the inequalities 
needed to show the tightness property needed in this proof.

The tightness property can also be proved in a different way.
Brown made it by means of Lemma~4 of his paper which follows
from different properties of martingales. An essential difference
between the formulation of our functional central limit theorem
and the corresponding result of Brown is that we formulated a
result about the behaviour of the random broken line process
$\tilde V_k(t)$ defined with the help of the set of (random) points 
$\zeta_{k,j}$, while Brown's result is about the behaviour of 
the random broken line $\tilde T_k(t)$ defined with the help of 
the set of (non-random) points $z_{k,j}$. The following
fact is behind this differences. Brown considered only such a
special case when the triangular arrays of martingale difference
sequences are defined with help of the normalization of a sequence
of martingale differences. In this case also the following stronger
version of formula~(1.2) holds:
$\limm_{k\to\infty}\frac{\zeta_{k,[kt]}}{z_{k,[kt]}}\Rightarrow1$
for all numbers $0< t\le 1$, where $[x]$ denotes the integer part
of the number $x$. This has the consequence that the (deterministic) 
points $z_{k,j}$ are very close to the (random) points $\zeta_{k,j}$, 
and the random broken lines $\tilde V_k(t)$ and $\tilde T_k(t)$ 
are close to each other for large indices~$k$. But in the general
case this statement does not hold any longer.

The above formulated functional central limit theorem deals not
with  the random broken line process $T_k(t)$, defined in a natural
way in formula~(3.7) from the partial sums of the martingale 
difference sequence $X_{k,j}$ with the help of the variances 
$d_{k,j}^2=EX_{k,j}^2$ of the individual terms. It deals with its
`randomly rescaled version' $V_k(t)$, which we get be replacing
the time points $z_{k,j}$ by the time points $\zeta_{k,j}$. It 
states about the processes $V_k(t)$ that it behaves for large
indices~$k$ similarly to a Wiener process. The analogous 
statement about $T_k(t)$ may not hold in some cases. We can say 
that the natural time scale of the process we are interested in 
is presented not by the partial sums of the variances $z_{k,j}$, 
but by the partial sums of the conditional variances 
$\zeta_{k,j}$. We can find a similar phenomenon in the behavior 
of stochastic processes $X(t)=\int_0^t f(s)W(\,ds)$ defined as 
the It\^o integral of a predictable stochastic process 
$f(\cdot)$. Such a stochastic process can be rescaled to a 
Wiener process by means of a (random) `inner time' of the 
process. (see the book of H.~P.~McKean Stochastic integrals, 
Section~2.5.) In more detail, if 
$X(t)=\int_0^t f(s,\oo)W(\,ds)$ is an It\^o integral, and we
introduce the `inner time of the process' 
$\tau(t)=\int_0^t f^2(s,\oo)\,ds$, then 
$Y(t)=X(\tau^{-1}(t))$ is a Wiener process (possibly stopped 
at a random time point).

\beginsection 4. Appendix: An application. L\'evy's 
characterization of Wiener processes.

We show, as an application of the central limit theorem for 
triangular arrays of martingale differences the proof of L\'evy's
characterization of Wiener processes. Let us recall, that a Wiener
process in the interval $[0,T]$, $0<T<\infty$, is such a Gaussian
stochastic process $W(t)$, $0\le t\le T$, for which $EW(t)=0$, 
$EW(s)W(t)=\min(s,t)$ for all parameters $0\le s,t\le T$, and 
its trajectories $W(\cdot,\oo)$ are continuous functions in 
the interval $[0,T]$ with probability~1. We shall deal with 
continuous time martingales. Let us recall also their definition. 

We say that a stochastic process $X(t)$, $0\le t\le T$, together with 
a class of increasing $\sigma$-algebras $\Cal F_t$, $0\le t\le T$,
(we say that a class of $\sigma$-algebras $\Cal F_t$ is increasing if
$\Cal F_s\subset\Cal F_t$ for $s\le t$) is a martingale if $X(t)$ is
$\Cal F_t$ measurable, $E|X(t)|<\infty$, and $E(X_t|\Cal F_s)=X_s$ 
with probability~1 if $0\le s\le t\le T$. We say that a stochastic 
process $X(t)$, $0\le t\le T$, is a martingale (without attaching 
a class of $\sigma$-algebras to it) if it is a martingale together 
with the class of $\sigma$-algebras $\Cal F_t$, $0\le t\le T$, 
defined as $\Cal F_t=\sigma(X(s),\; 0\le s\le t)$, $0\le t\le T$. 
We shall prove the following result.

\medskip\noindent
{\bf L\'evy's characterization of Wiener processes.} {\it A 
stochastic process $X(t)$, $0\le t\le T$, is a Wiener process 
in the interval $[0,T]$ if and only if it satisfies the 
following properties (a), (b) and (c).

\medskip
\item{(a)} $X(0)\equiv0$, and the process $X(t)$, $0\le t\le T$, 
is a martingale.
\item{(b)} The process $Y(t)=X^2(t)-t$, $0\le t\le T$, together 
with the $\sigma$-algebras $\Cal F_t=\sigma(X(s),\; 0\le s\le t)$, 
$0\le t\le T$, is a martingale.
\item{(c)} Almost all trajectories $X(\cdot,\oo)$ are continuous 
functions in the interval $[0,T]$.} 

\medskip\noindent
{\it Remark.} It is clear that a Wiener process satisfies 
conditions~(a), (b) and~(c). The following example shows that 
condition~(c) cannot be omitted from the conditions of the above 
result. Let $Z(t)$, $0\le t\le T$, be a Poisson process. Then 
$Z(t)-t$, $0\le t\le T$, satisfies conditions~(a) and~(b). 
Indeed, $Z(t)-t$  is a stochastic process with independent 
increments, and it is not difficult to check that it satisfies 
both properties~(a) and~(b). But this stochastic process, 
which is clearly not a Wiener process, does not satisfy 
property~(c). As we shall see in the proof of L\'evy's 
characterization of Wiener processes condition~(c) is closely 
related to the Lindeberg condition in the central limit theorem.

\medskip
We shall prove this result with the help of a lemma formulated
during the proof, and then we shall prove also this lemma.

\medskip\noindent
{\it The proof of L\'evy's characterization of Wiener processes
with the help of a lemma.} It is clear that a Wiener process 
satisfies conditions~(a), (b) and~(c). The hard part of the 
proof is to show that these conditions imply that $X(t)$ 
is a Wiener process. It is relatively simple to show with the 
help of properties~(a) and~(b) that the process $X(t)$ has 
expectation $EX(t)=0$ and covariance $EX(s)X(t)=\min(s,t)$, 
and we also assumed in condition~(c) that it has continuous 
trajectories. The hard part of the proof is to show that it 
is Gaussian, i.e. the finite dimensional distributions of
the process $X(t)$ are Gaussian. We shall use the central 
limit theorem for martingales to show this. We shall prove 
the following statement:
 
\medskip\noindent
If conditions (a), (b) and (c) hold, then for arbitrary positive
integer $k$, real numbers $u_1,\dots,u_k$ and  
$0\le t_1<t_2<\cdots<t_k\le T$ the random variable
$\summ_{j=1}^k u_j(X(t_j)-X(t_{j-1}))$ is normally distributed 
with expectation zero and variance 
$\summ_{j=1}^ku_j^2(t_j-t_{j-1})$. (Here we use the notation 
$t_0=0$.)

\medskip
By applying the above statement with fixed numbers
$0\le t_1<t_2<\cdots<t_k\le T$ for all real numbers 
$u_1,\dots,u_k$ we get that the random variables
$X(t_j)-X(t_{j-1})$, $1\le j\le k$, are independent
Gaussian random variables with expectation zero and
variance $t_j-t_{j-1}$. We can state this for all
sequences $0\le t_1<t_2<\cdots<t_k\le T$ this is 
equivalent to the statement that $X(t)$ is a Gaussian
process with expectation zero and covariance 
$EX(s)X(t)=\min(s,t)$. Hence it is enough to prove the 
above statement to complete the proof of the result. 

For the sake of simpler notations we shall prove this 
statement only in the special case $k=1$, $t_1=t$ and 
$u_1=1$. But it causes no problem to extend this proof 
to the general case.

A natural idea would be to apply the following method. Let us 
define for all $k=1,2,\dots$ the set of random variables
$X_{k,j}=\frac1{\sqrt t}[X(\frac {jt}k)-X(\frac{(j-1)t}k)]$ and 
the $\sigma$-algebras $\Cal F_{k,j}=\sigma(X_{k,1},\dots,X_{k,j})$,
$1\le j\le k$, and let $\Cal F_{k,0}$ be the trivial $\sigma$-algebra
$\Cal F_{k,0}=\{\emptyset,\Omega\}$. In such a way we defined a 
triangular array of martingale difference sequences (with $n_k=k$) 
that satisfies conditions (a) and (b) of the central limit theorem 
for triangular arrays of martingale differences. We still should 
show that it also satisfies condition~(c). We would like to exploit 
that almost all trajectories of the process $X(t)$ are continuous, 
hence uniformly continuous in the interval~$[0,T]$. This implies 
that for almost all $\oo\in\Omega$ there is a threshold index 
$k_0=k_0(\oo,\e)$ such that $|X_{k,j}(\oo)|<\e$ for all 
$1\le j\le k$ if $k\ge k_0$. Hence 
$\limm_{k\to\infty}\summ_{j=1}^{n_k}X_{k,j}^2I(|X_{k,j}|>\e)=0$
with probability~1. One would like to take expectation in this 
formula, which would lead to the Lindeberg formula~(1.1). But at 
this point some difficulty arises that we can overcome only by 
refining this argument with the help of a lemma. In this argument 
we work not directly with the random variable $X(t)$, but we 
approximate it with a sequence of random variables $X_n(t)$ for 
which we can apply the central limit theorem. To carry out this 
program we shall need the following lemma.

\medskip\noindent
{\bf Lemma.} {\it Let $X(t)$, $0\le t\le T$, be a stochastic 
process with continuous trajectories on the interval $[0,T]$. 
Let us fix two positive numbers $\e$ and $\alpha$, and define 
with their help the random variable $\tau$ as
$$
\align
\tau&=\tau(\e,\alpha,T)  \tag4.1 \\
&=\max\{t\colon\; t\le T,\quad |X(u)-X(v)|<\e
\quad \text{if } \; 0\le u\le v\le t, \text{ and }|u-v|\le\alpha\}.
\endalign
$$
This random variable $\tau=\tau(\e,\alpha,T)$ is a stopping 
time for all such class of increasing $\sigma$-algebras 
$\Cal F_t$, $0\le t\le T$, for which 
$\Cal B(X_s,\,0\le s\le t\}\subset \Cal F_t$. This stopping
time property means that 
$\{\oo\colon\;\tau(\oo)\le t\}\in \Cal F_t$ for all numbers 
$0\le t\le T$.

Let $g(x,u)$ be a continuous function on the set 
$[0,T]\times[0,\infty]$ for which the inequality 
$E|g(X(t),t)|<\infty$ holds for all $0\le t\le T$, and the 
stochastic process $g(X(t),t)$, $0\le t\le T$, together with 
some class of increasing class of $\sigma$-algebras 
$\Cal F_t$, $0\le t\le T$, is a martingale. Define, with the 
help of the previously defined stopping time $\tau$ the 
random  variables  $\tau_t=\min\{t,\tau\}$, $0\le t\le T$. 
Then the random variables $g(X(\tau_t),\tau_t)$, 
$0\le t\le T$ satisfy the identity 
$g(X(\tau_t),\tau_t)=E(g(X(T),T)|\Cal F_{\tau_t})$ with 
probability~1 for all $0\le t\le T$. Here $\Cal F_{\tau_t}$ 
consists  of those sets $B$ for which 
$B\cap\{\tau\le u\}\in\Cal F_u$ for all $0\le u\le T$.}

\medskip
We shall need the following corollary of the lemma.

\medskip\noindent
{\bf Corollary.} {\it If $X(s)$, $0\le s\le T$, is a 
stochastic process with continuous trajectories such 
that $EX^2(s)<\infty$ for all $0\le s\le T$, and the 
stochastic processes $X(s)$ and $Y(s)=X^2(s)-s$, 
$0\le s\le T$, are martingales together with some
increasing class of $\sigma$-algebras $\Cal F_s$,
$0\le s\le T$, then also the random processes $X(\tau_s)$
and $Y(\tau_s)=X^2(\tau_s)-\tau_s$  (with the stopping times 
$\tau_s$ defined in~(4.1), only with the notation of 
parameter~$t$ instead of parameter~$s$) are martingales
together with the $\sigma$-algebras $\Cal F_{\tau_s}$
in the interval $0\le s\le T$.}

\medskip
To prove L\'evy's characterization of Wiener processes
with the help of the corollary of the lemma let us consider 
the stochastic process $X(t)$, $0\le t\le s$, and introduce 
the stopping times $\tau^k=\tau(\e_k,\alpha_k,t)$ defined 
in formula~(4.1) with  the choice $\e_k=\frac1k$, $T=t$ and 
such an $\alpha_k>0$ for which
$$
P\(\supp_{0\le s,t\le 1,\;|t-s|\le\alpha_k}
|X(t,\oo)-X(s,\oo)|\ge\sqrt t\e_k\)\le\frac1{k^2}.
$$
Such an $\alpha_k$ exists because of the continuity of the
trajectories of the stochastic process $X(s)$, $0\le s\le t$. 
With such a choice $P(\tau^k=t)\ge 1-\frac1{k^2}$. Hence the
 random variables $X_k(t)=\frac1{\sqrt t}X(\tau^k_t)$
with $\tau^k_s=\min(s,\tau^k)$ for all $0\le s\le t$ converge
to $\frac1{\sqrt t}X(t)$ with probability~1, and to show that
$X(t)$ is normally distributed with expectation zero and 
variance~$t$  it is enough to prove that the random variables 
$X_k(t)$ converge in distribution to the standard normal 
distribution. We shall prove this with the help of the central
limit theorem for triangular arrays of martingale difference 
sequences. 

Let us choose for all $k=1,2,\dots$ some integer 
$n_k\ge\frac t{\alpha_k}$, and define the random variables
$X_{k,j}=\frac1{\sqrt t}[X(\tau^k_{\frac {jt}{n_k}})
-X(\tau^k_{\frac{(j-1)t}{n_k}})]$, $j=1,2,\dots,n_k$. Then we 
have $X_k(t)=\summ_{j=1}^{n_k} X_{k,j}$, and I claim that the
central limit theorem for triangular arrays of martingale
differences can be applied for the triangular array $X_{k,j}$,
$k=1,2,\dots$, $1\le j\le n_k$, and as a consequence the above
representation of $X_k(t)$ implies that the random 
variables $X_k(t)$ converge in distribution to the central
limit theorem. 

Indeed, the random variables $X_{k,j}$, $k=1,2,\dots$, 
$1\le j\le n_k$, constitute a triangular array of martingale 
difference sequences with the $\sigma$-algebras
$\Cal F_{j,k}=\Cal F_{\tau^k_{\frac {jt}{n_k}}}$, and this 
was property~(a) of this limit theorem. This corollary also 
implies that
$$ \allowdisplaybreaks
\align
E(X_{k,j}^2|\Cal F_{k,j-1})&=\frac1t \left.
E\(\(X\(\tau^k_{\frac{jt}{n_k}}\)
-X\(\tau^k_{\frac{(j-1)t}{n_k}}\)\)^2\right|\Cal F_{k,j-1}\)\\
&=\frac1t\biggl[ \left. E\(X\(\tau^k_{\frac {jt}{n_k}}\)^2
-\tau^k_{\frac{jt}{n_k}}\right|\Cal F_{k,j-1}\)
+\left.E\(\tau^k_{\frac{jt}{n_k}}\right|\Cal F_{k,j-1}\)\\
&\qquad -2X\(\tau^k_{\frac{(j-1)t}{n_k}}\)
\left.E\(X\(\tau^k_{\frac{jt}{n_k}}\)\right|\Cal F_{k,j-1}\)+
X\(\tau^k_{\frac{(j-1)t}{n_k}}\)^2\biggr]\\
&=\frac1t\biggl[X\(\tau^k_{\frac{(j-1)t}{n_k}}\)^2-\tau^k_{\frac{(j-1)t}{n_k}}
+\left.E\(\tau^k_{\frac{jt}{n_k}}\right|\Cal F_{k,j-1}\) \\
&\qquad\qquad -2X\(\tau^k_{\frac{(j-1)t}{n_k}}\)^2
+X\(\tau^k_{\frac{(j-1)t}{n_k}}\)^2\biggr]\\
&=\frac1t \[\left. E\(\tau^k_{\frac{jt}{n_k}}\right|\Cal F_{k,j-1}\)
-\tau^k_{\frac{(j-1)t}{n_k}}\]
=\frac1t \left. E\(\tau^k_{\frac{jt}{n_k}}-\tau^k_{\frac{(j-1)t}{n_k}}
\right|\Cal F_{k,j-1}\).
\endalign
$$
Observe that
$\frac1t \(\tau^k_{\frac{jt}{n_k}}-\tau^k_{\frac{(j-1)t}{n_k}}\)=
\frac1t\[\min\(\frac{jt}{n_k},\tau^k\)-\min\(\frac{(j-1)t}{n_k},\tau^k\)\]
\le\frac1{n_k}$,
and
$$
\frac1t \summ_{j=1}^{n_k}\(\tau^k_{\frac{jt}{n_k}}
-\tau^k_{\frac{(j-1)t}{n_k}}\)=\frac{\tau^k}t.
$$ 
We also know that $P(\tau^k=t)\to1$ as $k\to\infty$. This implies 
that the sequence  
$\frac1t \summ_{j=1}^{n_k}\(\tau^k_{\frac{jt}{n_k}}
-\tau^k_{\frac{(j-1)t}{n_k}}\)$ converges to~1 in $L_1$-norm as 
$k\to\infty$, and the same relation holds for the sequence
$\summ_{j=1}^{n_k}\sigma^2_{j,k}=\frac1t \summ_{j=1}^{n_k}
E\left.\(\tau^k_{\frac{jt}{n_k}}
-\tau^k_{\frac{(j-1)t}{n_k}}\right|\Cal F_{k,j-1}\)$. 
Hence property (b) also holds. 

Finally, by our construction 
$\summ_{j=1}^{n_k} X_{k,j}^2I(|X_{k,j}|\ge\e)\equiv0$ if $k\ge k_0(\e)$,
hence relation~(1.1) and thus condition (c) also holds. To complete
the proof it is enough to prove the lemma and its corollary.

\medskip\noindent
{\it Proof of the Lemma.}\/ Given a number $0\le t<T$, let
$Q_t$ denote the set of rational numbers in the interval
 $[0,t]$. Then $\{\tau\le T\}=\Omega\in\Cal F_T$, and
$$
\{\oo\colon\; \tau(\oo)\le t\}
=\bigcapp_{m=1}^\infty
\bigcupp_{(u,v)\colon\; u\in Q_t,\,v\in Q_t,\,|u-v|\le\alpha}
\left\{\oo\colon\; |X(u,\oo)-X(v,\oo)|\ge \e-\frac1m\right\}
\in \Cal F_t,
$$
for all $t<T$. This implies that $\tau(\oo)$ is a stopping 
time.  The identity in this relation holds, since $\tau(\oo)\le t$ 
if and only if there exist two such numbers
$0\le \bar u,\bar v\le t$ for which $|\bar u-\bar v|\le\alpha$,
and $|X(\bar u,\oo)-X(\bar v,\oo)|\ge\e$. On the other hand,
this relation holds if and only if for all $m=1,2,\dots$ there
are two numbers $u\in Q_t$ and $v\in Q_t$ such that 
$|u-v|\le \alpha$, and $X(u,\oo)-X(v,\oo)|\ge \e-\frac1m$. 
Indeed, if this relation hold, then there is a pair
$0\le\bar u\le \bar v\le t$ such that $|\bar u-\bar v|\le\alpha$,
and $|X(\bar u,\oo)-X(\bar v,\oo)|\ge\e$. Then we get by choosing
a sequence of pairs $u_n\in Q_t$ and $v_n\in Q_t$
such that $u_n\to \bar u$ and $v_n\to\bar v$ as $n\to\infty$,
that because of the continuity of the trajectories 
$|X(u_n,\oo)-X(v_n,\oo)|\ge\e-\frac1m$ for large indices~$n$. 
On the other hand, if the other statement holds, then we can
choose a sequence of pairs $u_n\in Q_t$ and $v_n\in Q_t$ such 
that $|u_n-v_n|\le\alpha$, and
$|X(u_n,\oo)-X(v_n,\oo)|\ge\e-\frac1n$ for large indices~$n$. 
By taking a subsequence $(u_{n_k}, v_{n_k})$ of these pairs
such that both sequences $u_{n_k}$ and $v_{n_k}$ are convergent
we get in the limit a pair $(\bar u,\bar v)$ in the limit
for which $|\bar u-\bar v|\le\alpha$, and
$|X(\bar u,\oo)-X(\bar v,\oo)|\ge\e$. Thus we proved that $\tau$
is a stopping time.

We shall prove the second statement of the lemma with the help of 
an appropriate discrete time approximation of the stochastic 
process $Z_u=g(X(u),u)$, $0\le u\le T$. We take for all positive 
integers $m=1,2,\dots$ and real numbers $0\le t\le T$ the random 
variables $Z_{\frac{jT}m}=g\(X(\frac{jT}m),\frac{jT}m\)$, the
$\sigma$-algebras $\Cal G_{\frac{jT}m}=\Cal F_{\frac{jT}m}$ 
$1\le j\le m$ together with the discretized stopping time 
$\tau_{t^{(m)}}$, which is defined by the formula  
$\tau_t^{(m)}=\frac{l}m$ if $\frac {l-1}m<\tau_t\le \frac{l}m$, 
$1\le l\le m$. (I would remark that it follows from the continuity
of the trajectories of the stochastic process $X(t)$ and the 
definition of the stopping time $\tau$ that $\tau(\oo)>0$ for 
almost all $\oo$.)

If $(g(X(u),u),\Cal F_u)$, $0\le u\le T$, is a martingale, then
$\(Z_{\frac{jT}m},\Cal G_{\frac {jT}m}\)$, $1\le j\le m$, is
also a martingale, and $\tau_t^{(m)}$ is a stopping time for it.
Hence it follows from a classical result for martingales that
$Z_{\tau_t^{(m)}}=E(Z_T|\Cal F_{\tau_{t^{(m)}}})$ with 
probability~1 for all integers $m=1,2,\dots$. On the other hand,
$Z_{\tau_t}=\limm_{m\to\infty}Z_{\tau_{t^{(m)}}}$ with 
probability~1 because of the continuity of the stochastic 
process $g((X(u,\oo),u)$. Hence to prove the identity we are 
working with it is enough to show that the random variables
$Z_{\tau_t^{(m)}}=E(Z_T|\Cal F_{\tau_{t^{(m)}}})$ converge to the
random variable $E(Z_T|\Cal F_{\tau_t})$ with probability~1 if
$m\to\infty$.

Moreover, to prove this statement it is enough to check that
the random variables
$Z_{\tau_t^{(m)}}=E(Z_T|\Cal F_{\tau_{t^{(m)}}})$, $m=1,2,\dots$,
are uniformly integrable. Indeed, we have to show that
$\int_A Z_T(\oo)\,dP=\int_A Z_{\tau_t}(\oo)\,dP$ for all sets
$A\in\Cal F_{\tau_t}$. On the other hand, we know that
$\int_A Z_T(\oo)\,dP=\int_A Z_{\tau_t^{(m)}}\,dP$ for all such
sets~$A$, and $Z_{\tau_t}=\limm_{m\to\infty}Z_{\tau_{t^{(m)}}}$ 
with probability~1. Hence the uniform integrability of the random
variables $Z_{\tau_t^{(m)}}$, $m=1,2,\dots$, enables us to carry
out a limiting procedure leading to the desired identity.

To prove the uniform integrability, (i.e. the inequality
$\int_{|Z_{\tau^{(m)}}|>K}|Z_{\tau^{(m)}}|\,dP\le\e$ for all 
numbers  $m=1,2,\dots$ if $K\ge K(\e)$ with a sufficiently
large number $K(\e)$) it is enough to show the following two
inequalities: (1.) For all numbers $\e>0$ there exists a number 
$\delta=\delta(\e)>0$ such that $\int_B |Z_{\tau^{(m)}}|\,dP\le\e$ 
if $P(B)\le\delta$ and $B\in\Cal F_{\tau_{t^{(m)}}}$, and (2.)
$P(|Z_{\tau^{(m)}}|>K)\le\delta$ for all numbers $m=1,2,\dots$ 
if $K\le K(\delta)$ with an appropriate number $K(\delta)$. The
first statement holds, because under our conditions
$\int_B |Z_{\tau^{(m)}}|\,dP\le\int_B |Z_T|\,dP$, and 
$\int_B |Z_T|\,dP<\e$, if $P(B)<\delta$. The second inequality
holds, because
$E|Z_{\tau^{(m)}}|=E|E(Z_T|\Cal F_{\tau_{t^{(m)}}})|)\le E|Z_T|$, 
and this implies that 
$P(|Z_{\tau^{(m)}}|>K)\le\frac{E|Z_{\tau^{(m)}}|}K\le
\frac{E|Z_T|}K\le\delta$ it $K\ge K(\delta)$. Thus we proved the
lemma.

\medskip\noindent
{\it Proof of the corollary of the lemma.}\/ To prove the
corollary of the lemma let us observe that under its conditions 
$X(\tau_s,\oo)=E(X(T,\oo)|\Cal F_{\tau_s})$, and
$X^2(\tau_{s},\oo)-\tau_s(\oo)=E(X^2(T,\oo)-T|\Cal F_{\tau_s})$. 
Hence, if $0\le s\le t\le T$, then 
$\Cal F_{\tau_s}\subset\Cal F_{\tau_t}$, and 
$$
E(X(\tau_t,\oo)|\Cal F_{\tau_s})
=E(E(X(T,\oo)|\Cal F_{\tau_t})|\Cal F_{\tau_s})
=E(X(T,\oo)|\Cal F_{\tau_s})=X(\tau_s,\oo).
$$ 
This is the first statement of the corollary. The second 
statement can be proved in the same way. 


\bye

