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Displaying similar documents to “On sums of i.i.d. random variables indexed by N parameters”

A recurrence theorem for square-integrable martingales

Gerold Alsmeyer (1994)

Studia Mathematica

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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...