Stochastic processes with random contexts: a characterization, and adaptive estimators for the transition probabilities

TL;DR

This paper introduces random context representations for finite-alphabet stochastic processes, generalizing context tree models, and develops an adaptive estimator that achieves near-optimal performance for certain renewal processes.

Contribution

It characterizes processes with random context representations, proves existence and uniqueness of minimal representations, and proposes an estimator with strong adaptivity and minimax performance.

Findings

Random context representations coincide with almost surely continuous transition probabilities.

Existence and uniqueness of minimal random context representations are established.

The proposed estimator achieves near-minimax performance for binary renewal processes.

Abstract

This paper introduces the concept of random context representations for the transition probabilities of a finite-alphabet stochastic process. Processes with these representations generalize context tree processes (a.k.a. variable length Markov chains), and are proven to coincide with processes whose transition probabilities are almost surely continuous functions of the (infinite) past. This is similar to a classical result by Kalikow about continuous transition probabilities. Existence and uniqueness of a minimal random context representation are proven, and an estimator of the transition probabilities based on this representation is shown to have very good "pastwise adaptativity" properties. In particular, it achieves minimax performance, up to logarithmic factors, for binary renewal processes with bounded $2+\gamma$ moments.