## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 3 (1992), 1204-1212.

### A Note on the Convergence of Sums of Independent Random Variables

#### Abstract

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 $\lim\inf_{n \rightarrow\infty}P(X_n = 0) > 0$.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 3 (1992), 1204-1212.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989687

**Digital Object Identifier**

doi:10.1214/aop/1176989687

**Mathematical Reviews number (MathSciNet)**

MR1175258

**Zentralblatt MATH identifier**

0762.60019

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 11K65: Arithmetic functions [See also 11Nxx]

**Keywords**

Probabilistic number theory additive arithmetic function limit distribution sums of independent random variables three series theorem

#### Citation

Hildebrand, Adolf. A Note on the Convergence of Sums of Independent Random Variables. Ann. Probab. 20 (1992), no. 3, 1204--1212. doi:10.1214/aop/1176989687. https://projecteuclid.org/euclid.aop/1176989687