Abstract
Full Text
UDC 519.21
MATHEMATICS
WERNER WOLF
SOME LIMIT THEOREMS FOR LARGE DEVIATIONS
(Presented by Academician Yu. V. Linnik on 12 VII 1967)
1. Consider a sequence of independent random variables \(X_1, X_2, \ldots\) with finite variances \(\sigma_1^2, \sigma_2^2, \ldots\), not all of which are equal to zero, and such that \(EX_j = 0\) \((j = 1, 2, \ldots)\). Put
\[ B_n^2=\sum_{j=1}^{n}\sigma_j^2,\qquad Z_n=\frac{1}{B_n}\sum_{j=1}^{n}X_j, \]
\[ F_n(x)=P\{Z_n<x\},\qquad \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}\,dt. \]
In the case of identical distributions, Cramér \((^1)\) obtained a theorem on the probabilities of large deviations of the quantity \(Z_n\), which was generalized by V. V. Petrov \((^2)\) to the case of non-identically distributed variables, with a simultaneous strengthening of it also for the particular case of identical distributions. We give the statement of Cramér’s theorem, taking this strengthening into account.
If \(X_1, X_2, \ldots\) is a sequence of independent identically distributed random variables and if, for some \(a>0\),
\[ Ee^{a|X_1|}<\infty, \tag{1} \]
then, for \(x>1\), \(x=o(\sqrt n)\), and \(n\to\infty\), we have
\[ \frac{1-F_n(x)}{1-\Phi(x)} = \exp\left\{\frac{x^3}{\sqrt n}\lambda\left(\frac{x}{\sqrt n}\right)\right\} \left[1+O\left(\frac{x}{\sqrt n}\right)\right], \]
\[ \frac{F_n(-x)}{\Phi(-x)} = \exp\left\{-\frac{x^3}{\sqrt n}\lambda\left(-\frac{x}{\sqrt n}\right)\right\} \left[1+O\left(\frac{x}{\sqrt n}\right)\right], \]
where
\[ \lambda(t)=c_0+c_1t+c_2t^2+\cdots \tag{2} \]
is a power series (Cramér’s series), convergent for all sufficiently small \(|t|\).
Analogous results under assumption (1) were obtained by W. Richter \((^3)\) for local theorems.
Yu. V. Linnik \((^{4-7})\) and V. V. Petrov \((^8,\,^9)\) obtained a number of integral and local limit theorems for the case where Cramér’s condition (1) is not fulfilled.
2. If \(s\) is a nonnegative integer, then \(\lambda_n^{[s]}(t)\) will denote the segment of the series \(\lambda_n(t)\) consisting of its first \(s+1\) terms:
\[ \lambda_n^{[s]}(t)=\sum_{k=0}^{s}c_{kn}t^k. \]
The series \(\lambda_n(t)\) is defined in the work of V. V. Petrov \((^{10})\). In the particular case of identical distributions this series coincides with Cramér’s series \(\lambda(t)\).
Theorem 1. Suppose that, for some \(\alpha,\ 0<\alpha<1/2\), the conditions
\[ \overline{\lim}_{n\to\infty}\frac1n\sum_{j=1}^n E\exp\{|X_j|^{4\alpha/(2\alpha+1)}\}<\infty, \tag{3} \]
\[ \overline{\lim}_{n\to\infty}\frac{B_n^2}{n}>0. \tag{4} \]
are satisfied. Then
\[ \frac{1-F_n(x)}{1-\Phi(x)} = \exp\left\{\frac{x^3}{\sqrt n}\lambda_n^{[s]}\left(\frac{x}{\sqrt n}\right)\right\} \left[1+O\left(\frac{x+1}{\sqrt n}\right)\right], \tag{5} \]
\[ \frac{F_n(-x)}{\Phi(-x)} = \exp\left\{-\frac{x^3}{\sqrt n}\lambda_n^{[s]}\left(-\frac{x}{\sqrt n}\right)\right\} \left[1+O\left(\frac{x+1}{\sqrt n}\right)\right] \tag{6} \]
as \(n\to\infty\) in the region \(0\le x\le n^\alpha/\rho(n)\), where \(\rho(n)\) is an arbitrary function satisfying the condition
\[ \lim_{n\to\infty}\rho(n)=+\infty . \tag{7} \]
Here \(s\) is a nonnegative integer determined by the inequalities
\[ s/2(s+2)<\alpha\le s+1/2(s+3). \tag{8} \]
We give one of the simplest corollaries of this theorem. Suppose that the conditions of Theorem 1 are satisfied for \(0<\alpha\le 1/6\). Then
\[ \lim_{n\to\infty}\frac{1-F_n(x)}{1-\Phi(x)}=1,\qquad \lim_{n\to\infty}\frac{F_n(-x)}{\Phi(-x)}=1 \]
in the region \(0\le x\le n^\alpha/\rho(n)\), whatever the function \(\rho(n)\) satisfying condition (7) may be.
Theorem 2. Suppose
\[ \overline{\lim}_{n\to\infty}\frac{B_n^2}{n}<\infty . \]
Suppose, further, that there exist positive constants \(\alpha<1/2,\ b_0\) and \(b_1\), and a function \(\rho(n)\) satisfying condition (7), such that
\[ 1-F_n(x)\le b_0e^{-b_1x^2},\qquad F_n(-x)\le b_0e^{-b_1x^2} \tag{9} \]
for \(0\le x\le n^\alpha\rho(n)\) and all sufficiently large \(n\). Then
\[ \overline{\lim}_{n\to\infty}\frac1n\sum_{j=1}^n E\exp\{|X_j|^{4\alpha/(2\alpha+1)}\}<\infty . \]
Let us note that inequalities (9) are satisfied in the region \(0\le x\le n^\alpha\rho(n)\) for a sufficiently slowly increasing function \(\rho(n)\), if relations (5) and (6) hold in this region and if the function \(\left|\lambda_n^{[s]}(x/\sqrt n)\right|\) is bounded.
- Introduce the notation
\[ v_j(t)=Ee^{itX_j},\qquad \varphi(x)=\frac1{\sqrt{2\pi}}e^{-x^2/2}. \]
Denote by \(p_n(x)\) the derivative of the distribution function \(F_n(x)\), if \(F_n(x)\) is absolutely continuous.
Theorem 3. Suppose that, for some \(\alpha,\ 0<\alpha<1/2\), the conditions
\[ \overline{\lim}_{n\to\infty}\frac1n\sum_{j=1}^n E\exp\{|X_j|^{4\alpha/(2\alpha+1)}\}<\infty, \tag{10} \]
\[ \overline{\lim}_{n\to\infty}\frac{B_n^2}{n}>0. \]
are satisfied. Suppose, further, that to every \(\varepsilon>0\) there corresponds such a \(\delta>0\) that
\[ \int_{|t|>\varepsilon}\prod_{j=1}^n |v_j(t)|\,dt = O\left(e^{-\delta n^{2\alpha}}\right) \qquad (n\to\infty). \]
Then, for all sufficiently large \(n\), there exists an everywhere continuous density \(p_n(x)\) of the distribution of the random variable \(Z_n\), and, moreover,
\[ \frac{p_n(x)}{\varphi(x)} = \exp\left\{ \frac{x^3}{\sqrt n}\, \lambda_n^{[s]}\!\left(\frac{x}{\sqrt n}\right) \right\} \left[ 1+O\!\left(\frac{|x|+1}{\sqrt n}\right) \right] \tag{11} \]
as \(n\to\infty\) in the region \(|x|\le n^\alpha/\rho(n)\), where \(\rho(n)\) is an arbitrary function satisfying condition (7), and \(s\) is the nonnegative integer determined by inequalities (8).
Corollary. Suppose the conditions of Theorem 3 are fulfilled for \(0<\alpha\le 1/6\). Then
\[ \lim_{n\to\infty}\frac{p_n(x)}{\varphi(x)}=1 \]
in the region \(|x|\le n^\alpha/\rho(n)\), whatever the function \(\rho(n)\) satisfying condition (7).
Theorem 4. Suppose
\[ \varlimsup_{n\to\infty}\frac{B_n^2}{n}<\infty \]
and the random variable \(Z_n\), for some \(n=n_0\), has an absolutely continuous distribution with density \(p_n(x)\). Suppose, furthermore, that there exist positive constants \(\alpha<1/2\), \(b_0\), and \(b_1\), and a function \(\rho(n)\) satisfying condition (7), such that
\[ p_n(x)\le b_0 e^{-b_1x^2} \tag{12} \]
for \(|x|\le n^\alpha\rho(n)\) and all sufficiently large \(n\). Then condition (3) is fulfilled.
- In papers \((^8,\ ^9)\) results close to Theorems 1 and 3 were obtained under the assumption that a condition stronger than (3) is fulfilled, namely,
\[ E\exp\left\{|X_j|^{4\alpha/(2\alpha+1)}\right\}\le C \qquad (j=1,2,\ldots), \tag{13} \]
where \(C\) is some constant. In addition, we have obtained more precise estimates of the remainder terms.
Theorems 2 and 4 are strengthenings of the corresponding theorems from \((^8,\ ^9)\). I express my deep gratitude to V. V. Petrov for the formulation of the problem and for valuable suggestions.
Leningrad State University
named after A. A. Zhdanov
Received
29 VI 1967
REFERENCES
- H. Cramér, Actual. sci. et ind., No. 736, Paris, 1938.
- V. V. Petrov, UMN, 9, no. 4, 195 (1954).
- V. V. Richter, Theory of Probability and Its Applications, 2, no. 2, 214 (1957).
- Yu. V. Linnik, DAN, 133, No. 6, 1291 (1960).
- Yu. V. Linnik, Theory of Probability and Its Applications, 6, 145, 377 (1961); 7, 121 (1962).
- Yu. V. Linnik, Proc. IV Berkeley Symp. on Math. Statistics and Probability, 2, 1961, p. 289.
- I. A. Ibragimov, Yu. V. Linnik, Independent and Stationarily Related Random Variables, Moscow, 1965.
- V. V. Petrov, Vestnik Leningrad Univ., No. 19, 49 (1963).
- V. V. Petrov, ibid., No. 1, 58 (1964).
- V. V. Petrov, ibid., No. 8, 13 (1953).