Different types of quasi weighted αβ-statistical convergence in probability

Abstract

The sequence of random variables {Xn}n ∈ ℕ is said to be weighted modulus αβ-statistically convergent in probability to a random variable X [16] if for any ε, δ > 0 (formula presented) where φ be a modulus function and {tn}n ∈ ℕ be a sequence of real numbers such that (formula presented) In this paper we study a related concept of convergence in which the value (formula presented) is replaced by (formula presented); for some sequence of real numbers {Cn}n ∈ ℕ such that (formula presented) (like [30]). The results are applied to build the probability distribution for quasi-weighted modulus αβ-statistical convergence in probability, quasi-weighted modulus αβ-strongly Cesàro convergence in probability, quasi-weighted modulus Sαβ-convergence in probability and quasiweighted modulus Nαβ-convergence in probability. If {Cn}n ∈ ℕ satisfying the condition (formula presented) then quasi-weighted modulus αβ-statistical convergence in probability and weighted modulus αβ-statistical convergence in probability are equivalent except the condition (formula presented). So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mention four convergences.