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February, 1984 Approximate Local Limit Theorems for Laws Outside Domains of Attraction
Philip S. Griffin, Naresh C. Jain, William E. Pruitt
Ann. Probab. 12(1): 45-63 (February, 1984). DOI: 10.1214/aop/1176993373


Let $\{X_k\}$ be a sequence of independent, identically distributed, nondegenerate random variables and $S_n = X_1 + \cdots + X_n$. Define $G(x) = P\{|X| > x\}, K(x) = x^{-2} \int_{|y| \leq x}y^2 dF(y), Q(x) = G(x) + K(x)$ for $x > 0$, and $\{a_n\}$ by $Q(a_n) = n^{-1}$ for large $n$. Let (A) denote the condition: $\lim \sup_{x \rightarrow \infty} G(x)/K(x) < \infty$. We show that (A) implies the following: there exist $\varepsilon > 0, C > 0$, such that for each $M > 0$ a sequence $\{b_n\}$ and a positive constant $c$ can be found for which $c \leqq a_nP\{S_n \in (x - \varepsilon, x + \varepsilon)\} \leq C$ whenever $|x - b_n| \leq Ma_n$ and $n$ is sufficiently large. In fact, the upper bound is valid for all $x$. We also prove that (A) is necessary for either the upper bound result or the lower bound result so that these results are equivalent. Feller had shown that (A) is equivalent to the existence of $\{\gamma_n\}, \{\delta_n\}$ such that the sequence $\{(S_n - \delta_n)/\gamma_n\}$ is stochastically compact.


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Philip S. Griffin. Naresh C. Jain. William E. Pruitt. "Approximate Local Limit Theorems for Laws Outside Domains of Attraction." Ann. Probab. 12 (1) 45 - 63, February, 1984.


Published: February, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0539.60022
MathSciNet: MR723729
Digital Object Identifier: 10.1214/aop/1176993373

Primary: 60F05
Secondary: 60G50

Keywords: local limit theorem , Random walk , stochastic compactness , tightness

Rights: Copyright © 1984 Institute of Mathematical Statistics


Vol.12 • No. 1 • February, 1984
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