Time-inhomogeneous random Markov chains. (9th October 2018) - Innocentini, G C P; Novaes, M
Abstract: We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, , with the factors M i drawn independently from an ensemble of random Markov matrices, whose columns are independent Dirichlet random variables. The statistical properties of the columns of, its largest eigenvalue and its spectrum are obtained exactly for N   =  2 and numerically investigated for general N . For large t, the columns are Dirichlet-distributed, however the distribution is different from the initial one. As for the spectrum, we find that the eigenvalues converge to zero exponentially fast and investigate the statistics of the largest stability and Lyapunov exponents, which are approximated by Gamma distributions. We also observe a concentration of the spectrum on the real line for large t .