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

Abstract

A sequence of real numbers {xn} n∈N is said to be αβ-statistically convergent of order γ (where 0 < γ ≤ 1) to a real number x [1] if for every δ > 0,where {αn} n∈N and {βn} n∈N be two sequences of positive real numbers such that {αn} n∈N and {βn} n∈N are both non-decreasing, βn ≥ αn ∀ n ∈ N, (βn −αn) → ∞ as n → ∞. In this paper we study a related concept of convergences in which the value |x k −x| is replaced by P (|X k −X| ≥ ε) and E(|X k − X| r ) repectively (Where X, X k are random variables for each k ∈ N, ε > 0, P denote the probability, E denote the expectation) and we call them αβ-statistical convergence of order γ in probability and αβ-statistical convergence of order γ in r th 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 behavior 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.Keywords: αβ-statistical convergence, αβ-statistical convergence of order γ in probability, αβ-strong p-Cesàro summability of order γ in probability, αβ-statistical convergence of order γ in r th expectation, αβ-statistical convergence of order γ in distribution. (2010) : 40A35, 40G15, 60B10. Mathematics Subject Classification

A Note on Probability Convergence Defined by Unbounded Modulus Function and $alphabeta$-Statistical Convergence

“…We mainly investigate their relationship and also make some observations about these classes. In the way we show that the notion of f − αβ-statistical convergence of order γ in probability is stronger than αβ-statistical convergence of order γ in probability (see [9]). It is important to note that the method of proofs and in particular examples are not analogous to the real case.Throughout the paper f will denote unbounded modulus function.…”

“…From Theorem 2.3 and Example 2.1 we see that this notion of f − αβ-statistical convergence of order γ in probability is stronger than αβ-statistical convergence of order γ in probability (see [9]).…”

“…In this case we write X n P S − − → X. The class of all sequences of random variables which are statistically convergent in probability is denoted by P S. One can also see [7,8,14] for related works.…”

Different types of quasi-weighted αβ-statistical convergence in probability

“…The class of all sequences of random variables which are statistically convergent in probability is denoted by (S, P). For more results on this convergence see the papers [9,17,18].…”

I2-lacunary strongly summability for multidimensional measurable functions

“…Active research on this topic started after the paper of Fridy [10] and since then a large collection of literature has appeared. At the last quarter of the 20th century, statistical convergence has been discussed and captured important aspect in creating the basis of several investigations conducted in main branches of mathematics such as the theory of number [7], measure theory [19], trigonometric series [31], probability theory [6], and approximation theory [12]. In addition, it was further investigated from the sequence space point of view and linked with summability theory by Connor [4], Et at.…”