## The Annals of Probability

### On the Local Limit Theorem for Independent Nonlattice Random Variables

Terence R. Shore

#### 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

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