A notion of $\alpha \beta$-statistical convergence of order $\gamma$ in probability

Abstract

A sequence of real numbers $\{x_n{\}}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma \le 1$) to a real number $x$ \cite{a} if for every $\delta >0,$ $$\underset{n\to \infty }{\lim }\frac{1}{({\beta }_n-{\alpha }_n+1{)}^{\gamma }}|\{k\in [{\alpha }_n,{\beta }_n]:|x_k-x|\ge \delta \}|=0.$$ where $\{{\alpha }_n{\}}_{n\in \mathbb{N}}$ and $\{{\beta }_n{\}}_{n\in \mathbb{N}}$ be two sequences of positive real numbers such that $\{{\alpha }_n{\}}_{n\in \mathbb{N}}$ and $\{{\beta }_n{\}}_{n\in \mathbb{N}}$ are both non-decreasing, ${\beta }_n\ge {\alpha }_n$ $\forall n\in \mathbb{N},$ (${\beta }_n-{\alpha }_n)\to \infty$ as $n\to \infty .$ In this paper we study a related concept of convergences in which the value $|x_k-x|$ is replaced by $P(|X_k-X|\ge \epsilon )$ and $E(|X_k-X{|}^r)$ repectively (Where $X,X_k$ are random variables for each $k\in \mathbb{N}$, $\epsilon >0$, $P$ denote the probability, $E$ denote the expectation) and we call them $\alpha \beta$-statistical convergence of order $\gamma$ in probability and $\alpha \beta$-statistical convergence of order $\gamma$ in $r^{\mbox{th}}$ expectation respectively. The results are applied to build the probability distribution for $\alpha \beta$-strong $p$-Ces$\grave{\mbox{a}}$ro summability of order $\gamma$ in probability and $\alpha \beta$-statistical convergence of order $\gamma$ in distribution. Our main objective is to interpret a relational behavior of above mentioned four convergences.

arXiv admin note: substantial text overlap with arXiv:1605.05555