On the Laplace transform of perpetuities with thin tails

Jean-Baptiste Bardet (1) , Hélène Guérin (2) , Florent Malrieu (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 IRMAR - Institut de Recherche Mathématique de Rennes

We consider the random variables $R$ which are solutions of the distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent of $R$ and $\ABS{M}\leq 1$. Goldie and Grübel showed that the tails of $R$ are no heavier than exponential. In this note we provide the exact lower and upper bounds of the domain of the Laplace transform of $R$.

Mots clés: Perpetuity; Stochastic difference equation; Laplace transform; tail behavior

Domaines: Probabilités [math.PR]

Jean-Baptiste Bardet, Hélène Guérin, Florent Malrieu. On the Laplace transform of perpetuities with thin tails. 2009. ⟨hal-00441640v2⟩