A recurrence theorem for square-integrable martingales

Let ${\displaystyle {\left(M_n\right)}_{n\ge 0}}$ be a zero-mean martingale with canonical filtration ${\displaystyle {\left({ℱ}_n\right)}_{n\ge 0}}$ and stochastically ${\displaystyle L_2}$-bounded increments ${\displaystyle Y_1,Y_2,...,}$ which means that ${\displaystyle P(\left|Y_n\right|>t\mid {ℱ}_{n-1})\le 1-H\left(t\right)}$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let ${\displaystyle V^2={\sum }_{n\ge 1}E\left(Y_n^2\mid {ℱ}_{n-1}\right)}$. It is the main result of this paper that each such martingale is a.s. convergent on {V < ∞} and recurrent on {V = ∞}, i.e. ${\displaystyle P(M_n\in [-c,c]i.\odot \mid V=\infty )=1}$ 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.