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Let ${\left(M_n\right)}_{n\ge 0}$ be a zero-mean martingale with canonical filtration ${\left({ℱ}_n\right)}_{n\ge 0}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,...,$ which means that $P\left(\right|Y_n|>t|{ℱ}_{n-1})\le 1-H\left(t\right)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2={\sum }_{n\ge 1}E\left(Y_n^2\right|{ℱ}_{n-1})$. 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....

Given any finite or countable collection of real numbers , ∈, we find all solutions to the stochastic fixed point equation $$W\stackrel{\mathrm{d}}{=}\underset{j\in J}{inf}{T}_j{W}_j,$$ where and the , ∈, are independent real-valued random variables with distribution and $\stackrel{\mathrm{d}}{=}$ 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...