## The Annals of Probability

- Ann. Probab.
- Volume 1, Number 4 (1973), 702-704.

### A Classical Limit Theorem Without Invariance or Reflection

#### Abstract

A sequence of stopping or first passage times is utilized to derive the limiting distribution of the maximum of partial sums of independent, identically distributed random variables with mean zero and finite variance and concomitantly the limit distribution of the stopping times themselves. The result, due to Erdos and Kac, first appeared in the paper which launched the extremely fruitful invariance principle; reflection enters in the calculations relating to the choice of a specific distribution for the $\{X_n\}$. Moreover, it is noted when the $\{X_n\}$ are $\operatorname{i.i.d.}$ with mean $\mu > 0$ and variance $\sigma^2 < \infty$ that $\max_{1\leqq j\leqq n} S_j/j^\alpha$ has a limiting standard normal distribution for any $\alpha$ in [0, 1).

#### Article information

**Source**

Ann. Probab., Volume 1, Number 4 (1973), 702-704.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996897

**Digital Object Identifier**

doi:10.1214/aop/1176996897

**Mathematical Reviews number (MathSciNet)**

MR350818

**Zentralblatt MATH identifier**

0262.60013

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G50: Sums of independent random variables; random walks

**Keywords**

First passage times invariance principles reflection maximum stable distribution stopping times positive normal distribution

#### Citation

Teicher, Henry. A Classical Limit Theorem Without Invariance or Reflection. Ann. Probab. 1 (1973), no. 4, 702--704. doi:10.1214/aop/1176996897. https://projecteuclid.org/euclid.aop/1176996897