The Annals of Probability

On the Local Limit Theorem for Independent Nonlattice Random Variables

Terence R. Shore

Full-text: Open access

Abstract

Let $(X_n: n \geqq 1)$ be a sequence of independent random variables, each having mean 0 and a finite variance. Under the Lindeberg condition and uniformity conditions on the characteristic functions, it is shown that the local limit theorem holds, i.e., if $S_n$ is the $n$th partial sum of the sequence, then $(2\pi \operatorname{Var} S_n)^{\frac{1}{2}}P(S_n \in (a, b)) \rightarrow b - a$. Under the assumption that the local limit theorem holds for each tail of $(X_n)$, and one other condition, it is then shown that the random walk generated by $(X_n)$ is recurrent if $\sum (\operatorname{Var} S_n)^{-\frac{1}{2}} = \infty$.

Article information

Source
Ann. Probab., Volume 6, Number 4 (1978), 563-573.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995478

Digital Object Identifier
doi:10.1214/aop/1176995478

Mathematical Reviews number (MathSciNet)
MR496406

Zentralblatt MATH identifier
0384.60016

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Local limit theorem random walk recurrence

Citation

Shore, Terence R. On the Local Limit Theorem for Independent Nonlattice Random Variables. Ann. Probab. 6 (1978), no. 4, 563--573. doi:10.1214/aop/1176995478. https://projecteuclid.org/euclid.aop/1176995478


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