## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 4 (1994), 1857-1878.

### On the Almost Sure Minimal Growth Rate of Partial Sum Maxima

Michael J. Klass and Cun-Hui Zhang

#### Abstract

Let $S_n = X_1 + \cdots + X_n$ be partial sums of independent identically distributed random variables and let $a_n$ be an increasing sequence of positive constants tending to $\infty$. This paper concerns the almost sure lower limit of $\max_{1\leq j \leq n} S_j/a_n$. We prove that the lower limit is either 0 or $\infty$ under mild conditions and give integral tests to determine which is the case. Let $\tau = \inf\{n \geq 1: S_n > 0\}$ and $\tau_- = \inf\{n \geq 1: S_n \leq 0\}$. Several inequalities are given that determine up to scale constants various quantities involving truncated moments of the ladder variables $S_\tau$ and $\tau$ under three different conditions: $ES_\tau < \infty, E|S_{\tau-}| < \infty$ and $X$ symmetric. Moments of ladder variables are also discussed.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 4 (1994), 1857-1878.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988487

**Digital Object Identifier**

doi:10.1214/aop/1176988487

**Mathematical Reviews number (MathSciNet)**

MR1331208

**Zentralblatt MATH identifier**

0857.60023

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60J15 60F15: Strong theorems

**Keywords**

Random walk integral test rate of escape ladder variable truncated moment inequality

#### Citation

Klass, Michael J.; Zhang, Cun-Hui. On the Almost Sure Minimal Growth Rate of Partial Sum Maxima. Ann. Probab. 22 (1994), no. 4, 1857--1878. doi:10.1214/aop/1176988487. https://projecteuclid.org/euclid.aop/1176988487