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A recurrence theorem for square-integrable martingales

Gerold Alsmeyer — 1994

Studia Mathematica

Let ( M n ) n 0 be a zero-mean martingale with canonical filtration ( n ) n 0 and stochastically L 2 -bounded increments Y 1 , Y 2 , . . . , which means that P ( | Y n | > t | 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 | 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....

A stochastic fixed point equation for weighted minima and maxima

Gerold AlsmeyerUwe Rösler — 2008

Annales de l'I.H.P. Probabilités et statistiques

Given any finite or countable collection of real numbers , ∈, we find all solutions to the stochastic fixed point equation W = d inf j J T j W j , where and the , ∈, are independent real-valued random variables with distribution and = d means equality in distribution. The bulk of the necessary analysis is spent on the case when ||≥2 and all are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting...

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