A stochastic fixed point equation for weighted minima and maxima

Abstract

Given any finite or countable collection of real numbers $T_j,j\in J$, we find all solutions $F$ to the stochastic fixed point equation \[W\stackrel{\mathrm {d}}{=}\inf_{j\in J}T_jW_j,\] where $W$ and the $W_j,j\in J$, are independent real-valued random variables with distribution $F$ and $\stackrel{\mathrm {d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when $|J|\geq 2$ and all $T_j$ are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation $T$ has a characteristic exponent $α$ given by $\sum_{j\in J}T_j^α=1$ and the set of solutions depends on the closed multiplicative subgroup of $\mathbb {R}^{>}=(0,\infty)$ generated by the $T_j$ which is either $\{1\}$, $\mathbb {R}^{>}$ itself or $r^{\mathbb {Z}}=\{r^n\dvt n\in \mathbb {Z}\}$ for some $r>1$. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to $\varLambda =\sum_{j\geq 1}δ_{T_j}$, i.e. $\varGamma =\varGamma \star \varLambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem.