Approximate Local Limit Theorems for Laws Outside Domains of Attraction

Abstract

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.