Abstract:

Let $\{X_n,n\geq1\}$ be a sequence of identically distributed
$\tilde{\rho}$ mixing random variables and $S_n=\sum\limits^n_{i=1}X_i(n\geq1)$. The paper discusses the upper bound of the distributions of $\max\limits_{1\leq i\leq n}\frac{|S_i|}{i}(n\geq1)$, and the sufficient and necessary conditions of the 1-$th$ and $p$-$th(p>1)$ moments of $\sup\limits_{n\geq1}\frac{|S_n|}{n}$ are obtained, which are as same as the case of the independent identically distributions.

Key words: ρ mixing sequence, Maximal inequality, The moment of
supremum function.