A recurrence theorem for square-integrable martingales
Volume 110 / 1994
Abstract
Let $(M_n)_{n≥0}$ be a zero-mean martingale with canonical filtration $(ℱ_n)_{n≥0}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,..., $ which means that $P(|Y_n| > t | ℱ_{n-1}) ≤ 1 - H(t)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2 = ∑_{n≥1} E(Y_{n}^{2}|ℱ_{n-1})$. It is the main result of this paper that each such martingale is a.s. convergent on {V < ∞} and recurrent on {V = ∞}, i.e. $P(M_{n} ∈ [-c,c] i.o. | V = ∞) = 1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions. As an application of our recurrence theorem, we obtain an extension of Blackwell's renewal theorem to a fairly general class of processes with independent increments and linear positive drift function.
