## The Annals of Probability

### Lower Class Sequences for the Skorohod-Strassen Approximation Scheme

David G. Kostka

#### Abstract

Let $S_n = X_1 + \cdots + X_n$ where $\{X_k\}_{k \geqq 1}$ is a sequence of independent, identically distributed random variables with mean zero and variance one. By the Skorohod representation $S_n$ has the same distribution as $\chi(U_n)$ where $\chi$ is standard Brownian motion. We find increasing sequences of real numbers $\{c_n\}$ and $\{d_n\}$ such that $$\lim \sum_{n\rightarrow\infty} \frac{\chi(U_n) - \chi(n)}{c_n \operatorname{lg} n)^{\frac{1}{2}}} = \infty \text{a.s}$$ and $$\lim \sup_{n\rightarrow\infty} \frac{\chi(U_n) - \chi(n)}{(d_n \operatorname{lg} n)^{\frac{1}{2}}} = 0 \text{a.s.}$$ We conclude with an example which explicitly gives the sequences $\{c_n\}$ and $\{d_n\}$ in terms of original random variables $\{X_k\}$.

#### Article information

Source
Ann. Probab., Volume 2, Number 6 (1974), 1172-1178.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996505

Digital Object Identifier
doi:10.1214/aop/1176996505

Mathematical Reviews number (MathSciNet)
MR358930

Zentralblatt MATH identifier
0294.60044

JSTOR
links.jstor.org

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G17: Sample path properties

#### Citation

Kostka, David G. Lower Class Sequences for the Skorohod-Strassen Approximation Scheme. Ann. Probab. 2 (1974), no. 6, 1172--1178. doi:10.1214/aop/1176996505. https://projecteuclid.org/euclid.aop/1176996505