General Erased-Word Processes: Product-Type Filtrations, Ergodic Laws and Martin Boundaries

TL;DR

This paper introduces General Erased-Word Processes (GEWPs), exploring their representation, boundary behavior, and filtration properties, revealing deep connections between exchangeability, order statistics, and product-type filtrations in stochastic processes.

Contribution

It provides a de Finetti-type representation for GEWPs, links exchangeability with poly-adic filtrations, and generalizes existing results on ergodic processes.

Findings

GEWPs have a de Finetti-type representation involving order statistics.

Ergodic GEWPs generate product-type filtrations.

The work generalizes Laurent's result on backward filtrations.

Abstract

We study the dynamics of erasing randomly chosen letters from words by introducing a certain class of discrete-time stochastic processes, general erased-word processes(GEWPs), and investigating three closely related topics: Representation, Martin boundary and filtration theory. We use de Finetti's theorem and the random exchangeable linear order to obtain a de Finetti-type representation of GEWPs involving induced order statistics. Our studies expose connections between exchangeability theory and certain poly-adic filtrations that can be found in other exchangeable random objects as well. We show that ergodic GEWPs generate backward filtrations of product-type and by that generalize a result by S.Laurent.