MATHEMATICS

A. V. SKOROKHOD

A LIMIT THEOREM FOR INDEPENDENT RANDOM VARIABLES

(Presented by Academician A. N. Kolmogorov on 16 IV 1960)

Let \(\xi_1, \xi_2, \ldots, \xi_n, \ldots\) be a sequence of independent identically distributed random variables for which \(M\xi_i=0,\ D\xi_i=1\). Put

\[ S_{n0}=0,\qquad S_{nk}=\frac{1}{\sqrt n}\sum_{i=1}^{k}\xi_i . \]

Further, let the functions \(g_1(t)\) and \(g_2(t)\) be defined for \(t\in[0,1]\), with \(g_1(0)<0<g_2(0)\), and for all \(t\in[0,1]\) let \(g_1(t)<g_2(t)\), and for some \(K\), for all \(t_1\) and \(t_2\) in \([0,1]\), the inequality

\[ |g_1(t_1)-g_1(t_2)|+|g_2(t_1)-g_2(t_2)|\le K|t_1-t_2| \]

holds.

Denote by \(Q_n\) the probability

\[ Q_n=P\left\{g_1\left(\frac{k}{n}\right)<S_{nk}<g_2\left(\frac{k}{n}\right),\quad k=0,1,2,\ldots,n\right\}. \]

Now consider the Brownian motion process \(w(t)\), for which \(Mw(t)=0,\ Dw(t)=t\), and denote

\[ Q=P\{g_1(t)<w(t)<g_2(t),\quad 0\le t\le 1\}. \]

It is known (see, for example, (1), Ch. IV, § 2) that \(Q_n\to Q\) as \(n\to\infty\). Under certain additional conditions one can estimate the rate of convergence of \(Q_n\) to \(Q\). However, even in the case of bounded \(\xi_i\), i.e., under the condition that for some \(c\)

\[ P\{|\xi_i|>c\}=0, \tag{*} \]

up to the present time one could use only the estimate of Yu. V. Prokhorov \((^2)\)

\[ |Q_n-Q|=O(n^{-1/4}\log^2 n). \]

We prove, under assumption (*), the stronger result:

Theorem. There exists a constant \(H\), depending only on \(K,\ C,\ g_1(0)\), and \(g_2(0)\), such that for all \(n\)

\[ |Q-Q_n|\le H\frac{\log n}{\sqrt n}. \tag{1} \]

This theorem is also valid in the case where \(\xi_1,\xi_2,\ldots\) have different distributions, but for all \(i\)

\[ M\xi_i=0,\qquad D\xi_i=1,\qquad P\{|\xi_i|>C\}=0. \]

The proof of the theorem is based on the following lemma, which may prove useful in studying the convergence of a sequence of sums of independent random variables to a Brownian-motion process.

Lemma. If \(\xi_1, \xi_2, \ldots, \xi_n\) are independent random variables for which \(\mathbf{M}\xi_i=0\), then one can specify independent nonnegative quantities \(\tau_1,\tau_2,\ldots,\tau_n\) such that the quantities

\[ w(\tau_1),\quad w(\tau_1+\tau_2),\quad w(\tau_1+\tau_2+\cdots+\tau_n) \]

(\(w(t)\) is a Brownian-motion process) have the same joint distribution as the quantities \(\xi_1,\xi_1+\xi_2,\xi_1+\xi_2+\cdots+\xi_n\), and moreover:

a) if \(\mathbf{D}\xi_i<\infty\), then \(\mathbf{M}\tau_i=\mathbf{D}\xi_i\);

b) there exist constants \(L_m,\ m>0\), such that

\[ \mathbf{M}\tau_i^m \leqslant L_m \mathbf{M}|\xi_i|^{2m}; \]

c) if \(|\xi_i|\leqslant C\), then

\[ \sup_{0\leqslant s\leqslant \tau_i} \left| w\left(\sum_{k=1}^{i-1}\tau_k+s\right) - w\left(\sum_{k=1}^{i-1}\tau_k\right) \right| \leqslant C; \]

d) if the \(\xi_i\) are identically distributed, then the \(\tau_i\) are also identically distributed.

Using this lemma, instead of the quantities \(s_{nk}\) we may consider the quantities \(w\left(\sum_{i=1}^{k}\tau_i^{(n)}\right)\), where \(\tau_i^{(n)}\) are the quantities corresponding by the lemma to the quantities \(\frac{1}{\sqrt n}\xi_i\). In this case \(\mathbf{M}\tau_i^{(n)}=0\), \(\mathbf{D}\tau_i^{(n)}=b^2/n^2\), where \(b^2\) is a certain constant.

Putting

\[ \frac{1}{\sqrt{n\mathbf{D}\tau_1^{(n)}}} \sum_{i=1}^{k}\left(\tau_i^{(n)}-\mathbf{M}\tau_i^{(n)}\right) = \frac{\sqrt n}{b} \left(\sum_{i=1}^{k}\tau_i^{(n)}-\frac{k}{n}\right) = \zeta_{nk}, \]

we shall have

\[ w\left(\sum_{i=1}^{k}\tau_i^{(n)}\right) = w\left(\frac{k}{n}+\frac{b}{\sqrt n}\zeta_{nk}\right). \]

Thus,

\[ Q_n= \mathbf{P}\left\{ g_1\left(\frac{k}{n}\right) < w\left(\frac{k}{n}+\frac{b}{\sqrt n}\zeta_{nk}\right) < g_2\left(\frac{k}{n}\right), \ k=0,1,2,\ldots,n \right\}. \]

Using this representation, part c) of the lemma and the fact that

\[ \mathbf{P}\left\{\sup_k|\zeta_{nk}|>\log n\right\}\leqslant \frac{1}{n^2}, \]

one can establish (1).

Received
12 IV 1960

References

1. A. Ya. Khinchin, Asymptotic Laws of Probability Theory, Moscow, 1936.
2. Yu. V. Prokhorov, Theory Probab. Appl., 1, 177 (1956).