2021/05/20 by Aleksandr A. Shchegolev, Shchegolev, Aleksandr, Aleksandr Shchegolev
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #FOS: Mathematics #Gene Regulatory Network Analysis #Markov Chains and Monte Carlo Methods #Petri Nets in System Modeling #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2105.09677
openalex publication_date 2021/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper studies an improved estimate for the rate of convergence for\nnonlinear homogeneous discrete-time Markov chains. These processes are\nnonlinear in terms of the distribution law. Hence, the transition kernels are\ndependent on the current probability distributions of the process apart from\nbeing dependent on the current state. Such processes often act as limits for\nlarge-scale systems of dependent Markov chains with interaction. The paper\ngeneralizes the convergence results by taking the estimate over two steps. Such\nan approach keeps the existence and uniqueness results under assumptions that\nare analogical to the one-step result. It is shown that such an approach may\nlead to a better rate of convergence. Several examples provided illustrating\nthe fact that the suggested estimate may have a better rate of convergence than\nthe original one. Also, it is shown that the new estimate may even be\napplicable in some cases when the conditions of the result on one step cannot\nguarantee any convergence. Finally, these examples depict that the original\nconditions may not be an obstacle for the convergence of nonlinear Markov\nchains.\n