A NOTION OF αβ-STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY

Abstract

A sequence of real numbers {x(n)}(n is an element of N) is said to be alpha beta-statistically convergent of order gamma (where 0 0, lim(n ->infinity) 1/(beta(n) - alpha(n) + 1)gamma vertical bar{k is an element of [alpha(n), beta(n)] : vertical bar x(k) - x vertical bar >= delta}vertical bar = 0, where {alpha(n)}(n is an element of N) and {beta(n)}(n is an element of N) are two sequences of positive real numbers such that {alpha(n)}(n is an element of N) and {beta(n)}(n is an element of N) are both non-decreasing, beta(n) >= alpha(n) for all n is an element of N, (beta(n) - alpha(n)) -> infinity as n -> infinity. In this paper we study a related concept of convergences in which the value x(k) is replaced by P (vertical bar X-k - X vertical bar >= epsilon) and E (vertical bar X-k - X vertical bar(r)) respectively (where X; X-k are random variables for each k is an element of N, epsilon > 0, P denotes the probability, and E denotes the expectation) and we call them alpha beta-statistical convergence of order gamma in probability and alpha beta-statistical convergence of order gamma in rth expectation respectively. The results are applied to build the probability distribution for alpha beta-strong p-Cesaro summability of order gamma in probability and alpha beta-statistical convergence of order gamma 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 (alpha, beta) sequences and this is also use to prove that if this condition violates then the limit value of alpha beta-statistical convergence of order gamma in probability of a sequence of random variables for two different (alpha, beta) sequences may not be equal.