The Expected Number of Intersections of a Four Valued Bounded Martingale with any Level May be Infinite

Abstract

According to the well-known Doob’s lemma, the expected number of crossings of every fixed interval (a,b) by trajectories of a bounded martingale (X _n ) is finite on the infinite time interval. For such a random sequence (r.s.) with an extra condition that X _n takes no more than N, N<∞, values at each moment n≥1, this result was refined in Sonin (Stochastics 21:231–250, 1987) by proving that inside any interval (a,b) there are non-random sequences (barriers) (d _n ), such that the expected number of intersections of d _n by (X _n ) is finite on the infinite time interval. This result left open the problem of whether for such r.s. any constant barriers d _n ≡d, n≥1, exist. The main result of this paper is an example of a bounded martingale X _n , 0≤X _n ≤1, with at most four values at each moment n, such that no constant d, 0<d<1, is a barrier for (X _n ). We also discuss the relationship of this problem with such problems as the behavior of a general finite nonhomogeneous Markov chain and the behavior of the simplest model of an irreversible process.