Discrete stochastic semigroups and associated difference equations with random coefficients (Q2777833)

Let \(\{X_n\), \(n\geq 1\}\) be a sequence of \(2\times 2\) random matrices with terms from the field of complex numbers \(\mathbb C\) and let \(Y_n=X_nX_{n-1}\cdots X_2X_1\), \(n\geq 1,\) be a sequence of products provided the factors are independent and have the Bernoulli distribution: for every \(n>1\) we have \(P\{X_n=A\}=p\), \(P\{X_n=B\}=q\), \(p+q=l\), \(0<p<1,\) \(A\) and \(B\) are elements of the space \(L(\mathbb C^2)\). The asymptotic behavior of the sequence \(Y_n\) is characterized by the Lyapunov exponent NEWLINE\[NEWLINE\kappa=\lim_{n\to\infty}n^{-1}\ln\|Y_n\|= \lim_{n\to\infty}n^{-1}{\mathbf E}\ln\|Y_n\|\pmod P,NEWLINE\]NEWLINE where \(\|\cdot\|\) is an arbitrary norm in \(L(\mathbb C^2)\). The author shows that investigation of the asymptotic behavior of \(Y_n\) may be reduced to investigation of the asymptotic behavior of solutions of the corresponding difference equations with random coefficients.