V. V. PETROV

ON INTEGRAL THEOREMS FOR LARGE DEVIATIONS

(Presented by Academician A. N. Kolmogorov on 24 I 1961)

In paper (¹) a number of new local limit theorems for large deviations were obtained. In the present paper we give the corresponding integral limit theorems. Throughout this note we shall use the notation of paper (¹), namely, the notation of Section 1 of that paper for the case of identical distributions and the notation of Section 2 for the case of nonidentical distributions.

1. Let (X_1, X_2) be a sequence of independent identically distributed random variables with finite variance (\sigma^2>0) and expectation (EX_1) equal to zero. Put

[
F_n(x)=\mathbf P\left{\frac{X_1+X_2+\cdots+X_n}{\sigma\sqrt n}<x\right}, \qquad
\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-t^2/2}\,dt .
\tag{1}
]

Theorem 1. If

[
E\exp |X_1|^{\frac{4\alpha}{2\alpha+1}}<\infty
\tag{2}
]

for some (\alpha) ((0<\alpha<1/2)), then for (0\le x\le n^\alpha/\rho(n)), where (\rho(n)) is an arbitrary function such that (\lim\limits_{n\to\infty}\rho(n)=+\infty), as (n\to\infty) we have

[
1-F_n(x)=[1-\Phi(x)]\exp\left{\frac{x^3}{\sqrt n}\lambda^{[s]}\left(\frac{x}{\sqrt n}\right)\right}[1+o(1)],
\tag{3}
]

[
F_n(-x)=\Phi(-x)\exp\left{-\frac{x^3}{\sqrt n}\lambda^{[s]}\left(-\frac{x}{\sqrt n}\right)\right}[1+o(1)]
\tag{3a}
]

uniformly with respect to (x). Here (s) is a nonnegative integer determined by the inequalities

[
\frac{s}{2(s+3)}<\alpha\le \frac{s+1}{2(s+3)},
\tag{4}
]

and (\lambda^{[s]}(t)) is the segment of Cramér’s series consisting of its first (s) terms.

Condition (2) is necessary in order that relations (3) and (3a) hold for some integer (s\ge0), (0\le x\le n^\alpha\rho(n)), and (n\to\infty) (even if not uniformly with respect to (x)), where (\alpha) is some constant and (\rho(n)) is some function such that (0<\alpha<1/2), (\lim\limits_{n\to\infty}\rho(n)=+\infty).

The following very general assertion is also true:

Theorem 2. Suppose that for (0\le x\le n^\alpha\rho(n)) and all sufficiently large values of (n) the inequalities

[
1-F_n(x)\le c_0 e^{-c_1x^2}, \qquad
F_n(-x)\le c_0 e^{-c_1x^2},
]

hold, where (\rho(n)) is some function satisfying the condition (\lim\limits_{n\to\infty}\rho(n)=+\infty); (\alpha), (c_0), and (c_1) are some positive constants, with (\alpha<1/2). Then condition (2) is satisfied for the given (\alpha).

From the assertions formulated one can obtain theorems concerning conditions for normal convergence (2). Let (\rho(n)) denote a function increasing (arbitrarily slowly) to infinity, and let (\gamma_m) denote the cumulant (semi-invariant) of order (m) of the random variable (X_1).

Corollary 1. Let (0<\alpha<{}^{1}/_6). Condition (2) is sufficient in order that, for (0\leq x\leq n^\alpha/\rho(n)) and (n\to\infty), the relations
[
\frac{1-F_n(x)}{1-\Phi(x)}\to 1,\qquad
\frac{F_n(-x)}{\Phi(-x)}\to 1
\tag{5}
]
hold uniformly with respect to (x), and is necessary in order that relations (5) hold for (0\leq x\leq n^\alpha\rho(n)) and (n\to\infty).

Corollary 2. Let (\alpha) satisfy condition (4) for some integer (s>0). Condition (2) and
[
\gamma_m=0,\qquad (m=3,4,\ldots,s+2)
\tag{6}
]
are sufficient in order that relations (5) hold for (0\leq x\leq n^\alpha/\rho(n)) and (n\to\infty) uniformly with respect to (x), and are necessary in order that (5) be fulfilled for (0\leq x\leq n^\alpha\rho(n)) and (n\to\infty).

1. Consider a sequence of independent, generally speaking, not identically distributed random variables (X_1, X_2,\ldots) with mathematical expectations equal to zero. Denote by (F_n(x)) the distribution function of the normalized sum of the random variables (X_1,X_2,\ldots,X_n).

Theorem 3. Let (0<\alpha\leq{}^{1}/6); (\rho(n)) be an arbitrary function satisfying the condition (\lim\rho(n)=+\infty). If
[
\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n} E X_j^2>0,\qquad
E\exp |X_j|^{\frac{4\alpha}{2\alpha+1}}\leq C
\quad (j=1,2,\ldots),
]
where (C) is some constant, then for (0\leq x\leq n^\alpha/\rho(n)) and (n\to\infty) relations (5) hold uniformly with respect to (x).

Theorem 4. If for the sequence of random variables under consideration the condition
[
\varlimsup_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n} E X_j^2<\infty
]
is fulfilled, and if the distribution function (F_n(x)) satisfies the conditions of Theorem 2, then
[
E\exp |X_j|^{\frac{4\alpha}{2\alpha+1}}<\infty
]
for all (j) ((j=1,2,\ldots)).

Leningrad State University
named after A. A. Zhdanov

Received
8 I 1961

CITED LITERATURE

¹ V. V. Petrov, DAN, 134, No. 3, 525 (1960). ² Yu. V. Linnik, DAN, 133, No. 6, 1291 (1960).