Statistical $p$-convergence in lattice-normed Riesz spaces

Abstract

A sequence $(x_n)$ in a lattice-normed space $(X,p,E)$ is statistical $p$-convergent to $x\in X$ if there exists a statistical $p$-decreasing sequence $q\stpd 0$ with an index set $K$ such that $\delta(K)=1$ and $p(x_{n_k}-x)\leq q_{n_k}$ for every $n_k\in K$. This convergence has been investigated recently for $(X,p,E)=(E,|\cdot|,E)$ under the name of statistical order convergence and under the name of statistical multiplicative order convergence, and also, for taking $E$ as a locally solid Riesz space under the names statistically unbounded $\tau$-convergence and statistically multiplicative convergence. In this paper, we study the general properties of statistical $p$-convergence.