Characterizations of strong and statistical convergences

Abstract

The two main objectives of the paper are to cast the concept of Auniform integrability in the measure-theoretic sense leading to showing that a sequence is A-strongly convergent if and only if it is A-statistically convergent and it is A-uniformly integrable. The second aim is to give an almost all subsequence characterization of A-statistical convergence.

Modulated Lacunary Statistical and Strong-Cesàro Convergences

“…If a sequence (x n ) is bounded and statistically convergent then (x n ) is strong-Cesàro convergent; this is the result of Connor [6]. Later, Khan and Orhan [24] sharpened this result by proving that a sequence (x n ) is statistically convergent if and only if (x n ) is strong-Cesàro convergent and uniformly integrable.…”