A recurrence theorem for square-integrable martingales

Alsmeyer, Gerold

Studia Mathematica, Tome 108 (1994), p. 221-234 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ${(M_{n})}_{n \geq 0}$ be a zero-mean martingale with canonical filtration ${(ℱ_{n})}_{n \geq 0}$ and stochastically $L_{2}$ -bounded increments $Y_{1}, Y_{2}, . . .,$ which means that $P (| Y_{n} | > t | ℱ_{n - 1}) \leq 1 - H (t)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^{2} = \sum_{n \geq 1} E (Y_{n}^{2} | ℱ_{n - 1})$ . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P (M_{n} \in [- c, c] i . o . | V = \infty) = 1$ 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.

Publié le : 1994-01-01

Zbl 0806.60073

EUDML-ID : urn:eudml:doc:216110

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     title = {A recurrence theorem for square-integrable martingales},
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     volume = {108},
     year = {1994},
     pages = {221-234},
     zbl = {0806.60073},
     language = {en},
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Alsmeyer, Gerold. A recurrence theorem for square-integrable martingales. Studia Mathematica, Tome 108 (1994) pp. 221-234. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv110i3p221bwm/

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