UDC 519.21

MATHEMATICS

L. V. OSIPOV

ON AN ASYMPTOTIC EXPANSION IN THE CENTRAL LIMIT THEOREM

(Presented by Academician Yu. V. Linnik on 16 IX 1965)

Let \(X_1, X_2,\ldots\) be a sequence of mutually independent identically distributed random variables with mathematical expectation \(m=EX_1\) and positive variance \(\sigma^2=E(X_1-m)^2\). Introduce the notation

\[ F_n(x)=P\left\{\frac{1}{\sigma\sqrt n}\sum_{j=1}^n (X_j-m)<x\right\},\quad \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-t^2/2}\,dt. \]

Theorem. Suppose that the following conditions are satisfied:

1) \(E|X_1|^k<\infty\) for some integer \(k\geq 3\);

2) \[ \lim_{|t|\to\infty}\left|Ee^{itX_1}\right|<1. \]

Then there exists a function \(\varepsilon(n)\), independent of \(x\), such that
\[ \lim_{n\to\infty}\varepsilon(n)=0 \]
and

\[ \left|F_n(x)-\Phi(x)-\sum_{\nu=1}^{k-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}}\right| \leq \frac{\varepsilon(n)}{(1+|x|^k)n^{(k-2)/2}} \]

for all \(x\) \((-\infty<x<\infty)\). The functions \(P_\nu(-\Phi)\) are defined in the same way as in \((^{1,4})\).

This theorem is a refinement of a theorem of Esseen \((^{1,2})\), according to which, under the conditions of the theorem formulated above, the relation

\[ F_n(x)-\Phi(x)-\sum_{\nu=1}^{k-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}} = o\left(\frac{1}{n^{(k-2)/2}}\right) \]

holds as \(n\to\infty\), uniformly with respect to \(x\) \((-\infty<x<\infty)\).

V. V. Petrov \((^3)\) earlier found conditions necessary and sufficient for the validity of the relation

\[ \left| \frac{d}{dx}\left(F_n(x)-\Phi(x)-\sum_{\nu=1}^{k-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}}\right) \right| \leq \frac{\varepsilon(n)}{(1+|x|^k)n^{(k-2)/2}} \]

for all \(x\) \((-\infty<x<\infty)\), where \(\varepsilon(n)\) does not depend on \(x\) and
\[ \lim_{n\to\infty}\varepsilon(n)=0. \]

From the theorem stated above there follows directly a series of consequences concerning the global form of integral limit theorems.

Under the conditions of the theorem, the following assertions are valid:

1. For any \(p>1/k\), as \(n\to\infty\) we have

\[ \int_{-\infty}^{\infty} \left| F_n(x)-\Phi(x)-\sum_{\nu=1}^{k-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}} \right|^p dx = o\left(\frac{1}{n^{(k-2)p/2}}\right). \]

1. For any \(p \geqslant 1\), as \(n \to \infty\) we have

\[ \int_{-\infty}^{\infty} |F_n(x)-\Phi(x)|^p dx = \int_{-\infty}^{\infty} \left| \sum_{\nu=1}^{k-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}} \right|^p dx + o\!\left(\frac{1}{n^{(k+p-3)/2}}\right). \]

1. For any \(p \geqslant 1\), as \(n \to \infty\) we have

\[ \|F_n(x)-\Phi(x)\| = \left\| \sum_{\nu=1}^{k-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}} \right\| + o\!\left(\frac{1}{n^{(k-2)/2}}\right). \]

Here

\[ \|u(x)\|=\left[\int_{-\infty}^{\infty}|u(x)|^p dx\right]^{1/p} \]

for any function \(u(x)\in L_p(-\infty,\infty)\).

Leningrad State University
named after A. A. Zhdanov

Received
7 IX 1965

REFERENCES

1. B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, 1949.
2. G.-G. Esseen, Acta Math., 77, 1 (1945).
3. V. V. Petrov, Theory of Probability and Its Applications, 9, 343 (1964).
4. V. V. Petrov, Vestnik Leningrad Univ., No. 19, 150 (1962).