Permanental sequences that are related to a Markov chain example of Kolmogorov

TL;DR

This paper investigates permanental sequences derived from Markov chain potentials, analyzing their divergence, continuity, and discontinuity behaviors at zero, with results depending on kernel parameters.

Contribution

It introduces a detailed analysis of non-symmetric permanental sequences related to Markov chains, providing exact rates and moduli of continuity at zero.

Findings

Exact divergence rates at zero depending on kernel parameters

Precise local modulus of continuity at zero

Characterization of bounded discontinuity at zero

Abstract

Permanental sequences with non-symmetric kernels that are generalization of the potentials of a Markov chain with state space $\{0,1/2, \ldots, 1/n,\ldots\}$ that was introduced by Kolmogorov, are studied. Depending on a parameter in the kernels we obtain an exact rate of divergence of the sequence at $0$, an exact local modulus of continuity of the sequence at $0$, or a precise bounded discontinuity for the sequence at $0$.