General erased-word processes: Product-type filtrations, ergodic laws and Martin boundaries

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 [La16].2010 Mathematics Subject Classification. 52A07, 60G09, 60G99, 60J50.