Lower Class Sequences for the Skorohod-Strassen Approximation Scheme

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 $\{X_k\}$.