Different Types of Quasi Weighted αβ-Statistical Convergence in Probability

Pratulananda Das, Sanjoy Ghosal, Sumit Som


The sequence of random variables ${\{X_{n}}\}_{n\in \mathbb{N}}$ is said to be weighted modulus $\alpha\beta$-statistically convergent in probability to a random variable $X$ \cite{Ghosal2014(3)} if for any $\varepsilon, \delta>0,$
$$\underset{n\rightarrow \infty}{\lim}\frac{1}{T_{\alpha\beta(n)}}~ |\{k \leq T_{\alpha\beta(n)}: t_{k}\phi(P(|X_{k}-X|\geq \varepsilon))\geq \delta\}|=0,$$
where $\phi$ be a modulus function and $\{t_{n}\}_{n\in \mathbb{N}}$ be a sequence of  real numbers such that $\displaystyle{\liminf_{n\rightarrow \infty}} t_n>0$ and $T_{\alpha\beta(n)}=\displaystyle{\sum_{k\in [\alpha_{n},\beta_{n}]}} t_k $,  $\forall ~n\in \mathbb{N}.$ In this paper we study a related concept of convergence in which the value $\frac{1}{T_{\alpha\beta(n)}}$ is replaced by $\frac{1}{C_n},$ for some sequence of real numbers $\{C_n\}_{n\in \mathbb{N}}$ : $C_{n}>0,~ \forall~ n\in \mathbb{N},$ $\underset{n\rightarrow \infty}{\lim}C_n=\infty$ and $\underset{n\rightarrow \infty}{\limsup} \frac{C_n}{T_{\alpha\beta(n)}}<\infty$ (like \cite{Sakaoglu-Ozguc2012}). The results are applied to build the probability distribution for quasi-weighted modulus $\alpha \beta$-statistical convergence, quasi-weighted modulus $\alpha \beta$-strong Ces$\grave{\mbox{a}}$ro convergence, quasi-weighted modulus $S_{\alpha \beta}$-convergence and  quasi-weighted modulus $N_{\alpha \beta}$-convergence. If $\{C_{n}\}_{n\in \mathbb{N}}$ satisfying the condition  $\underset{n\rightarrow \infty}{\liminf} \frac{C_n}{T_{\alpha\beta(n)}}>0,$ then quasi-weighted modulus $\alpha\beta$-statistical convergence in probability and weighted modulus $\alpha\beta$-statistical convergence in probability are equivalent except the condition {\textbf{$\underset{n\rightarrow \infty}{\liminf} \frac{C_n}{T_{\alpha\beta(n)}}=0$}}. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mention four convergences.

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