On the Local Limit Theorem for Independent Nonlattice Random Variables

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 $\sum (\operatorname{Var} S_n)^{-\frac{1}{2}} = \infty$.