  Classical Banach principle is an essential tool for the investigation of the
ergodic properties of Cesaro subsequences. The aim of this work is to extend
Banach principle to the case of the stochastic convergence in the operator
algebras. We start by establishing a sufficient condition for the stochastic
convergence (stochastic Banach principle). Then we formulate stochastic
convergence for the bounded Besicovitch sequences, and, as consequence for
uniform subsequences.
