Markov Approximations of chains of infinite order in the $\bar{d}$-metric

TL;DR

This paper provides explicit bounds on the distance between infinite-order chains and their finite Markov approximations, using a constructive coupling method that applies to a broad class of processes.

Contribution

It introduces a new constructive approach to bound the $ar{d}$-distance for chains of infinite order, including non-continuous kernels and null transitions.

Findings

Derived explicit upper bounds for the $ar{d}$-distance.

Applicable to non-continuous probability kernels and chains with null transitions.

Proved the Bernoulli property for these processes.

Abstract

We derive explicit upper bounds for the $\bar{d}$-distance between a chain of infinite order and its canonical $k$-steps Markov approximation. Our proof is entirely constructive and involves a "coupling from the past" argument. The new method covers non necessarily continuous probability kernels, and chains with null transition probabilities. These results imply in particular the Bernoulli property for these processes.