2018/05/31 by G. C. P. Innocentini, G C P Innocentini, M. Novaes +1 · 5 citations
Business, Management and Accounting · Mathematics · Physics and Astronomy · #Advanced Queuing Theory Analysis #Dirichlet distribution #Discrete phase-type distribution #Eigenvalues and eigenvectors #Markov Chains and Monte Carlo Methods #Markov chain #Markov chain mixing time #Markov process #Markov property #Random Matrices and Applications #Random matrix #Spectrum (functional analysis) #Variable-order Markov model #math-ph #math.MP
paper · pdf · doi:10.1088/1742-5468/aae028
published in Journal of Statistical Mechanics Theory and Experiment 2018(10), 103202 (Institute of Physics)
openalex created_date 2018/06/01 · openalex publication_date 2018/10/01 · arxiv created 2018/10/19 · arxiv updated 2018/11/14 · openalex updated_date 2026/08/05
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 .