On a generalization of a theorem of W. Sudderth and some applications

A. Engelbert; H. Engelbert

Displaying similar documents to “On a generalization of a theorem of W. Sudderth and some applications”

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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...

A note on dilations and martingales.

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