A notion of $\alpha\beta$-statistical convergence of order $\gamma$ in probability
| Autor: | Das, Pratulananda, Ghosal, Sanjoy, Karakaya, Vatan, Som, Sumit |
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| Rok vydání: | 2016 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\delta>0,$ $$\underset{n\rightarrow \infty} {\lim} \frac{1}{(\beta_{n} - \alpha_{n} + 1)^\gamma}~ |\{k \in [\alpha_n,\beta_n] : |x_{k}-x|\geq \delta \}|=0.$$ 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, $\beta_{n}\geq \alpha_{n}$ $\forall ~n\in \mathbb{N},$ ($\beta_{n}-\alpha_{n})\rightarrow \infty$ as $n\rightarrow \infty.$ In this paper we study a related concept of convergences in which the value $|x_{k}-x|$ 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 $\alpha \beta$-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. Comment: arXiv admin note: substantial text overlap with arXiv:1605.05555 |
| Databáze: | arXiv |
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