Statistical convergence analogues of some matrix summability theorems

In 1943, R. C. Buck characterized convergence by proving that if x is a nonconvergent sequence, then no regular matrix can sum every subsequence of x. This result was extended in 1944 by R. P. Agnew who showed that given a regular matrix A and a bounded sequence x, there is a subsequence y of x such that the set of limit points of Ay includes the set of limit points of x. Analogues to these results were given by D. Dawson in 1973 and J. Fridy in 1975 by replacing subsequence with stretching and rearrangement, respectively. In this dissertation we give statistical convergence analogues to Buck's theorem and its variants; specifically, we show that a sequence x is convergent if and only if every subsequence (respectively, rearrangement, stretching) of x is statistically convergent. Using the notion of a statistical limit point, we establish statistical convergence analogues to Agnew's result by proving that every sequence x has a subsequence (respectively, rearrangement, stretching) y such that every limit point of x is a statistical limit point of y. We then extend our results to the more general A-statistical convergence, where A is an arbitrary nonnegative regular matrix. Finally, we describe a class of matrices for which A-statistical convergence is equivalent to ordinary convergence and a class of sequences that are not A-statistically convergent for any nonnegative regular matrix A.