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August, 1978 On the Local Limit Theorem for Independent Nonlattice Random Variables
Terence R. Shore
Ann. Probab. 6(4): 563-573 (August, 1978). DOI: 10.1214/aop/1176995478


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


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Terence R. Shore. "On the Local Limit Theorem for Independent Nonlattice Random Variables." Ann. Probab. 6 (4) 563 - 573, August, 1978.


Published: August, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0384.60016
MathSciNet: MR496406
Digital Object Identifier: 10.1214/aop/1176995478

Primary: 60F05
Secondary: 60G50

Keywords: local limit theorem , Random walk , recurrence

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • August, 1978
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