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July, 1992 A Note on the Convergence of Sums of Independent Random Variables
Adolf Hildebrand
Ann. Probab. 20(3): 1204-1212 (July, 1992). DOI: 10.1214/aop/1176989687


Let $X_n, n \geq 1$, be a sequence of independent random variables, and let $F_N$ be the distribution function of the partial sums $\sum^N_{n = 1}X_n$. Motivated by a conjecture of Erdos in probabilistic number theory, we investigate conditions under which the convergence of $F_N(x)$ at two points $x = x_1,x_2$ with different limit values already implies the weak convergence of the distributions $F_N$. We show that this is the case if $\sum^\infty_{n = 1}\rho(X_n,c_n) = \infty$ whenever $\sum^\infty_{n = 1}c_n$ diverges, where $\rho(X,c)$ denotes the Levy distance between $X$ and the constant random variable $c$. In particular, this condition is satisfied if $\lim\inf_{n \rightarrow\infty}P(X_n = 0) > 0$.


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Adolf Hildebrand. "A Note on the Convergence of Sums of Independent Random Variables." Ann. Probab. 20 (3) 1204 - 1212, July, 1992.


Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0762.60019
MathSciNet: MR1175258
Digital Object Identifier: 10.1214/aop/1176989687

Primary: 60F05
Secondary: 11K65

Keywords: Additive arithmetic function , limit distribution , Probabilistic number theory , Sums of independent random variables , three series theorem

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 3 • July, 1992
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