$(\sigma,\lambda)$-Asymptotically Statistical Equivalent Sequences

Rahmet Savaş, Metin Başarir

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent, $\lambda$-statistical convergence and $\sigma$-convergence. Two nonnegative sequences $[x]$ and $[y]$ are said to be $S_{\sigma,\lambda}$-asymptotically equivalent of multiple $L$ provided that for every $\epsilon>0$ \[ \underset{n}\lim\frac1\lambda_n\quad k\in I_n:\frac{x_{\sigma^k(m)}}{y_{\sigma^k(m)}}-L\geq\epsilon=0 \] uniformly in $m=1,2,3,\ldots$ (denoted by $x\overset{S_{\sigma,\lambda}}\sim y$) and simply $S_{\sigma,\lambda}$-asymptotically equivalent, if $L=1$. Using this definition we shall prove $S_{\sigma,\lambda}$-asymptotically equivalent analogues of Mursaleen's theorems in [8].