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).