Quenched phantom distribution functions for Markov chains

TL;DR

This paper investigates how to recover phantom distribution functions for positive Harris recurrent Markov chains, including Metropolis algorithms, by starting from a specific point rather than the stationary distribution, aiding the analysis of maxima behavior.

Contribution

It introduces a method to obtain phantom distribution functions for a broad class of Markov chains starting from a fixed point, extending previous approaches that relied on stationarity.

Findings

Phantom distribution functions can be recovered from a fixed starting point.

Applicable to a large class of positive Harris recurrent Markov chains.

Enhances understanding of maxima behavior in Markov processes.

Abstract

It is known that random walk Metropolis algorithms with heavy-tailed target densities can model atypical (slow) growth of maxima, which in general is exhibited by processes with the extremal index zero. The asymptotics of maxima of such sequences can be analyzed in terms of continuous phantom distribution functions. We show that in a large class of positive Harris recurrent Markov chains (containing the above Metropolis chains) a phantom distribution function can be recovered by starting "at the point" rather than from the stationary distribution.