Open Access
August, 1973 A Classical Limit Theorem Without Invariance or Reflection
Henry Teicher
Ann. Probab. 1(4): 702-704 (August, 1973). DOI: 10.1214/aop/1176996897


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).


Download Citation

Henry Teicher. "A Classical Limit Theorem Without Invariance or Reflection." Ann. Probab. 1 (4) 702 - 704, August, 1973.


Published: August, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0262.60013
MathSciNet: MR350818
Digital Object Identifier: 10.1214/aop/1176996897

Primary: 60F05
Secondary: 60G40 , 60G50

Keywords: first passage times , Invariance principles , Maximum , positive normal distribution , reflection , stable distribution , stopping times

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 4 • August, 1973
Back to Top