On convergence rate bounds for a class of nonlinear Markov chains
Alexander Shchegolev, Alexander Veretennikov
TL;DR
This work tackles the problem of deriving convergence-rate bounds for nonlinear Markov chains by replacing classical Markov–Dobrushin-type conditions with a spectral-radius framework inspired by linear Markov chains and small nonlinear perturbations. The authors introduce two core assumptions that enable a Markovian-coupling construction and define a linear skeleton operator $\bar{V}$ whose spectral radius $r(\bar{V})$ governs exponential ergodicity rates via bounds of the form $\|\mu_n - \nu_n\|_{TV} \le 2C\bigl(r(\bar{V})+\delta\bigr)^n$ for small perturbations. They also establish a limsup rate bound tied to $\log r(\bar{V})$ as perturbations vanish, and show that the new spectral-radius condition can yield stronger or more flexible results than traditional MD-type coefficients, with stronger non-asymptotic bounds when $\bar{V}$ is compact. The paper provides finite-state examples demonstrating that $r(\bar{V})$ can yield tighter convergence estimates than the corresponding $1-\alpha_k$ bounds, and discusses implications for Extreme Value theory via the $D$-condition. Overall, the approach offers a practical, theoretically sharper tool for assessing decorrelation speeds in nonlinear Markov dynamics.
Abstract
A new approach is developed for evaluating the convergence rate for nonlinear Markov chains (MC) based on the recently developed spectral radius technique of markovian coupling for linear MC and the idea of small nonlinear perturbations of linear MC. The method further enhances recent advances in the problem of convergence for such models. The new convergence rate may be used, in particular, for the justification of $D$-condition in the Extreme Values theory.