Statistical convergence in Lebesgue spaces is considered in this paper. A criterion for statistical convergence is given. It is shown that the known Tauberian theorems for scalar case are valid in this case, too.
"Statistical convergence of functional sequences." Rocky Mountain J. Math. 45 (5) 1413 - 1423, 2015. https://doi.org/10.1216/RMJ-2015-45-5-1413