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December, 1974 Lower Class Sequences for the Skorohod-Strassen Approximation Scheme
David G. Kostka
Ann. Probab. 2(6): 1172-1178 (December, 1974). DOI: 10.1214/aop/1176996505

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 $\{X_k\}$.

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David G. Kostka. "Lower Class Sequences for the Skorohod-Strassen Approximation Scheme." Ann. Probab. 2 (6) 1172 - 1178, December, 1974. https://doi.org/10.1214/aop/1176996505

Information

Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0294.60044
MathSciNet: MR358930
Digital Object Identifier: 10.1214/aop/1176996505

Subjects:
Primary: 60G50
Secondary: 60G17

Keywords: lower class sequences , Skorohod representation , upper class sequences

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 6 • December, 1974
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