Stationary entrance Markov chains, inducing, and level-crossings of random walks

Abstract

For an arbitrary (possibly transient) Markov chain Y with values in a Polish space, consider the entrance Markov chain obtained by sampling Y at the moments when it enters a fixed set A from its complement A c . Similarly, consider the exit Markov chain, obtained by sampling Y at the exit times from A c into A. This paper provides a framework for analysing invariant measures of these two types of Markov chains in the case when the initial chain Y has a known σ-finite invariant measure. Under certain mild assumptions we give explicit formulas for invariant measures of these chains. Then we study their uniqueness and ergodicity assuming that Y is topologically recurrent, irreducible, and weak Feller. Our approach is based on the technique of inducing from infinite ergodic theory.We give applications to random walks in R d , which we regard as "stationary" Markov chains started under the Lebesgue measure. We are mostly interested in dimension one, where we study the Markov chain of overshoots above the zero level of a random walk that oscillates between −∞ and +∞. We show that this chain is ergodic, and use this result to prove a central limit theorem for the number of level crossings for random walks with zero mean and finite variance of increments.

Stability of overshoots of zero mean random walks

“…Since this proof gives no insight into the form of π + , in Section 2.3 we present a heuristic argument which we used to find this invariant distribution. The invariance of π + is also established in our companion paper [15,Corollary to Theorem 3] in a much more general setting using entirely different methods based on infinite ergodic theory; the proof presented here precedes the one in [15]. By [15, Corollary to Theorem 4], the assumption in (1) implies that the law π + is a unique (up to multiplicative constant) locally finite Borel invariant measure of the chain of overshoots O on Z.…”

“….+S k ) 1≤k≤n stays positive (see Vysotsky [23,24]). A detailed discussion with applications and further connections to a special class of Markov chains called random walks with switch at zero, is available in [15,Section 1.2]. Let us mention that distributions of the same form as π + appear on many occasions -this is discussed in Sections 2.1 and 2.2.…”

Ergodic aspects of trading with threshold strategies

“…They also establish the convergence of Law(O n ) to µ * in an appropriate sense, depending on the exact assumptions. Generalizations to entrance Markov chains on more general state spaces have been obtained in [12].…”

“…where by Assumption 2.1, (12) holds true, and thus for any fixed u ∈ [0, 1], the random map x → Φ(x, u, η n+1 ) is constant on with probability α showing that X ′ n forgets its previous state with positive probability. This observation will play later a central role later.…”

“…We also investigate a closely related object, studied in [11,12]: the so-called overshoot process. We extend certain results from [11] from i.i.d.…”