On the tail distribution of the solution to some law equation

Abstract

We consider a distribution equation which was initially studied by Bertoin \cite{Bertoin}: \[M \stackrel{d}{=} \max\{\widetildeν, \max_{1\leq k\leq ν}M_k\}.\] where $\{M_k{\}}_{k\ge 1}$ are i.i.d. copies of $M$ and independent of $(\tilde{\nu },\nu )\in {\mathbb{R}}_{+}\times \mathbb{N}$. We obtain the tail behaviour of the solution of a generalised equation in a different but direct method by considering the joint tail of $(\tilde{\nu },\nu )$.

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