A survey of max-type recursive distributional equations

Abstract

In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form $X\mathop{=}\limits^{d}\,g((\xi_{i},X_{i}),i\geq 1)$. Here (ξ_i) and g(⋅) are given and the X_i are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(⋅) is essentially a “maximum” or “minimum” function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process X_i, are the X_i measurable functions of the innovations process (ξ_i)?