The expected number of intersections of a four valued bounded martingale with any level may be infinite

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 (Xn) is finite on the infinite time interval. For such a random sequence (r.s.) with an extra condition that Xn takes no more than N, N < ∞, values at each moment n ≥ 1, this result was refined in Sonin (1987) by proving that inside any interval (a,b) there are nonrandom sequences (barriers) (dn), such that the expected number of intersections of dn by (Xn) is finite on the infinite time interval. This result left open the problem of whether for such r.s. any constant barriers dn ≡ d, n≥ 1, exist. The main result of this paper is an example of a bounded martingale Xn, 0≤ Xn ≤ 1, with at most four values at each moment n, such that no constant d, 0 < d < 1, is a barrier for (Xn). 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.