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Let be a zero-mean martingale with canonical filtration and stochastically -bounded increments which means that a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. 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....
Given any finite or countable collection of real numbers
, ∈, we find all solutions to the stochastic fixed point equation
where and the
, ∈, are independent real-valued random variables with distribution and 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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