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

A sequence of real numbers $\{x_n{\}}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma\leq 1$)toarealnumber$x$\text{cite}aifforevery$\delta>0,$ $$\frac{1}{({\beta }_n-{\alpha }_n+1{)}^{\gamma }}\mathrm{\mid }\{k\in [{\alpha }_n,{\beta }_n]:\mathrm{\mid }{x}_k-x\mathrm{\mid }\ge \delta \}\mathrm{\mid }=0.$$ 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, $\beta_{n}\geq \alpha_{n}$ $\mathrm{\forall }n\in \mathbb{N},$ ($\beta_{n}-\alpha_{n})\rightarrow \infty$as$n\rightarrow \infty.$ In this paper we study a related concept of convergences in which the value $\mathrm{\mid }{x}_k-x\mathrm{\mid }$ 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 $k\in \mathbb{N}$, $\epsilon >0$, $P$ denote the probability, $E$ denote the expectation) and we call them $\alpha \beta$-statisticalconvergenceoforder$\gamma$inprobabilityand$\alpha\beta$-statisticalconvergenceoforder$\gamma$in$r^{\mbox{th}}$ expectation respectively. The results are applied to build the probability distribution for $\alpha \beta$-strong $p$-Ces$\grave{a}$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.