Time-inhomogeneous random Markov chains

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, $U(t)=M_t\cdots M_1$, 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 $U(t)$, 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 Lyapunov exponent, which is well approximated the a Gamma distribution. We also observe a concentration of the spectrum on the real line for large $t$.

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Cited by in corpus (5)

• On the number of real eigenvalues of a product of truncated orthogonal random matrices
• Spectral statistics for the difference of two Wishart matrices
• Singular value distribution of dense random matrices with block Markovian dependence
• Random stochastic matrices from classical compact Lie groups and symmetric spaces
• Statistics of the non-zero eigenvalues and singular values of low-rank random matrices with non-negative entries