TAUBERIAN THEOREMS IN THE STATISTICAL SENSE FOR THE WEIGHTED MEANS OF DOUBLE SEQUENCES

Let $p:=\{p_j\}_{j=0}^{\infty}$ and $q:=\{q_k\}_{k=0}^{\infty}$ be complex sequences with $p,q \in$ SVA. Assume that $\{s_{mn}\}_{m,n=0}^\infty$ is a double sequence in $\mathbf{C}$ (or one of $\mathbf{R}$, a Banach space, and an ordered linear space), with $s_{mn} \stackrel{st}{\rightarrow} s \hspace{6 pt} (\mbox{$\bar{\rm N}$},p,q;\alpha,\beta)$, where $(\alpha,\beta)=(1,1)$, $(1,0)$ or $(0,1)$. We give sufficient and/or necessary conditions under which $s_{mn} \stackrel{st}{\rightarrow} s$. The theory developed here is the statistical version of the work of Chen and Hsu in [{\em Anal. Math.}, {\bf 26} (2000), 243-262]. Our results generalize M\'oricz, [{\em J. Math. Anal. Appl.}, {\bf 286} (2003), 340-350].