A notion of αβ-statistical convergence of order γ in probability

Abstract

A sequence of real numbers {xn}n_N is said to be ɑβ-statistically convergent of order (where 0 < ᵞ ≤ 1) to a real number x [1] if for every b > 0, lim n͢1 1 (βn - ɑn + 1) |{k ∈ [ɑn, βn] : |xk - x| ≥ b}| = 0, where {ɑn}n2N and {βn}n2N are two sequences of positive real numbers such that {ɑn}n2N and {βn}n2N are both non-decreasing, βn ≥ ɑn for all n ∈ N, (βn -ɑn) → ∞ as n → ∞. In this paper we study a related concept of convergences in which the value xk is replaced by P(|Xk - X| ≥ e) and E(|Xk - X|r) respectively (where X,Xk are random variables for each k ∈ N, e > 0, P denotes the probability, and E denotes the expectation) and we call them ɑβ -statistical convergence of order in probability and ɑβ-statistical convergence of order in rth expectation respectively. The results are applied to build the probability distribution for ɑβ -strong p-Cesàro summability of order in probability and ɑβ -statistical convergence of order 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 (ɑ, β) sequences and this is also use to prove that if this condition violates then the limit value of ɑβ - statistical convergence of order in probability of a sequence of random variables for two different (ɑ, β) sequences may not be equal.