A Notion of $\alpha\beta$-statistical Convergence of Order $\gamma$ in Probability

Pratulananda Das ; Sumit Som ; Sanjoy Ghosal ; Vatan Karakaya

Kragujevac Journal of Mathematics, Tome 42 (2018) no. 1, p. 51

Résumé

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $\alpha\beta$-statistically convergent of order $\gamma$ (where $0\gamma\leq 1$) to a real number $x$ \cite{a} if for every $\delta>0$, $\underset{n\rightarrow ıfty}{im}\frac{1}{(\beta_{n}-lpha_{n}+1)^\gamma} eft|\{k ı [lpha_n,\beta_n] : |x_{k}-x|\geq ẹlta \}\right|=0,$ where $\{\alpha_{n}\}_{n\in \mathbb{N}}$ and $\{\beta_{n}\}_{n\in \mathbb{N}}$ are 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, $\beta_{n}\geq \alpha_{n}$ for all $n\in \mathbb{N}$, ($\beta_{n}-\alpha_{n})\rightarrow \infty$ as $n\rightarrow \infty$. In this paper we study a related concept of convergences in which the value $x_{k}$ is replaced by $P(|X_{k}-X|\geq \varepsilon)$ and $E(|X_{k}-X|^{r})$ respectively (where $X, X_k$ are random variables for each $k\in \mathbb{N}$, $\varepsilon>0$, $P$ denotes the probability, and $E$ denotes the expectation) and we call them $\alpha \beta$-statistical convergence of order $\gamma$ in probability and $\alpha\beta$-statistical convergence of order $\gamma$ in $r^{\text{th}}$ expectation respectively. The results are applied to build the probability distribution for $\alpha\beta$-strong $p$-Ces+�ro summability of order $\gamma$ in probability and $\alpha\beta$-statistical convergence of order $\gamma$ in distribution. So our main objective is to interpret a relational behaviour of above mentioned four convergences. We give a condition under which a sequence of random variables will converge to a unique limit under two different $(\alpha,\beta)$ sequences and this is also use to prove that if this condition violates then the limit value of $\alpha \beta$-statistical convergence of order $\gamma$ in probability of a sequence of random variables for two different $(\alpha,\beta)$ sequences may not be equal.

Détail

Classification : 40A35 40G15, 60B10
Keywords: $\alpha\beta$-statistical convergence, $\alpha \beta$-statistical convergence of order $\gamma$ in probability, $\alpha\beta$-strong $p$-Cesàro summability of order $\gamma$ in probability, $\alpha\beta$-statistical convergence of order $\gamma$ in $r^\textth$ expectation, $\alpha\beta$-statistical convergence of order $\gamma$ in distribution