Asymptotic behavior of a class of stochastic semigroups in the Bernoulli scheme

Abstract

The family of subalgebras that describe the space of complex-valued $2 \times 2$ matrices is selected. In this space, the stochastic semigroup $Y_n = X_n X_{n-1} ... X_1, \; n = \overline{1, \infty}$, is considered, where $\{X_ , і = \overline{1, \infty}\}$ are independent equally distributed random matrices taking two values. For the stochastic semigroup $Y_n$, whose phase space belongs to one of the subalgebras, the index of exponential growth is calculated explicitly.