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 $\{X_k\}$.
Publié le : 1974-12-14
Classification:
Skorohod representation,
lower class sequences,
upper class sequences,
60G50,
60G17
@article{1176996505,
author = {Kostka, David G.},
title = {Lower Class Sequences for the Skorohod-Strassen Approximation Scheme},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 1172-1178},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996505}
}
Kostka, David G. Lower Class Sequences for the Skorohod-Strassen Approximation Scheme. Ann. Probab., Tome 2 (1974) no. 6, pp. 1172-1178. http://gdmltest.u-ga.fr/item/1176996505/