arXiv subjects

Valery Korchevsky

Publications and source records attributed to Valery Korchevsky.

3 recordsLinked to original sources

arXiv2017

On the Strong Law of Large Numbers for Sequences of Pairwise Independent Random Variables

We establish new sufficient conditions for the applicability of the strong law of large numbers (SLLN) for sequences of pairwise independent non-identically distributed random variables. These results generalize Etemadi's extension of Kolmogorov's SLLN for identically distributed random variables. Some of the obtained results hold with an arbitrary norming sequence in place of the classical normalization.

math.PR↗

arXiv2015

Marcinkiewicz-Zygmund Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

It is shown that the Marcinkiewicz-Zygmund strong law of large numbers holds for pairwise independent identically distributed random variables. It is proved that if $X_{1}, X_{2}, \ldots$ are pairwise independent identically distributed random variables such that $E|X_{1}|^p < \infty$ for some $1 < p < 2$, then $(S_{n}-ES_{n})/n^{1/p} \to 0$ a.s. where $S_{n} = \sum_{k=1}^{n} X_{k}$.

math.PR↗

arXiv2014

A Generalization of the Petrov Strong Law of Large Numbers

In 1969 V.V.~Petrov found a new sufficient condition for the applicability of the strong law of large numbers to sequences of independent random variables. He proved the following theorem: let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of independent random variables with finite variances and let $S_{n}=\sum_{k=1}^{n} X_{k}$. If $Var (S_{n})=O (n^{2}/ψ(n))$ for a positive non-decreasing function $ψ(x)$ such that $\sum 1/(n ψ(n)) < \infty$ (Petrov's condition) then the relation $(S_{n}-ES_{n})/n \to 0$ a.s. holds. In 2008 V.V.~Petrov showed that under some additional assumptions Petrov's condition remains sufficient for the applicability of the strong law of large numbers to sequences of random variables without the independence condition. In the present work, we generalize Petrov's results (for both dependent and independent random variables), using an arbitrary norming sequence in place of the classical normalization.

math.PR↗