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


Das P., Som S., Ghosal S., KARAKAYA V.

Kragujevac Journal of Mathematics, vol.42, no.1, pp.51-67, 2018 (Refereed Journals of Other Institutions) identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 1
  • Publication Date: 2018
  • Doi Number: 10.5937/kgjmath1801051d
  • Title of Journal : Kragujevac Journal of Mathematics
  • Page Numbers: pp.51-67

Abstract

© Kragujevac Journal of Mathematics 2018.A sequence of real numbers [xn]n∈ℕ is said to be αβ-statistically convergent of order (where 0 < γ ≤ 1) to a real number x [1] if for every δ > 0, where [αn]n∈ℕ and [βn]n∈ℕ are two sequences of positive real numbers such that [αn]n∈ℕ and [βn]n∈ℕ are both non-decreasing, βn ≥ αn for all n ∈ ℕ, (βn ≥αn) → ∞ as n → ∞. In this paper we study a related concept of convergences in which the value xk is replaced by P(|Xk - X| ≥ ε) and E(|Xk - X|r) respectively (where X;Xk are random variables for each k ∈ ℕ, ε > 0, P denotes the probability, and E denotes the expectation) and we call them αβ-statistical convergence of order γ in probability and αβ-statistical convergence of order γ in rth expectation respectively. The results are applied to build the probability distribution for αβ-strong p-Cesàro summability of order γ in probability and αβ-statistical convergence of order γ 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 (α, β) sequences and this is also use to prove that if this condition violates then the limit value of αβ- statistical convergence of order γ in probability of a sequence of random variables for two different (α, β) sequences may not be equal.