A notion of $αβ$-statistical convergence of order $γ$ in probability

Abstract

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