On tails of perpetuities

TL;DR

This paper derives an upper bound on the tail distribution of solutions to stochastic difference equations, aligning with previous lower bounds in the non-negative case, thus advancing understanding of tail behaviors in such stochastic processes.

Contribution

It provides a new upper bound on the tails of perpetuities, complementing existing lower bounds and enhancing theoretical understanding of their distributional properties.

Findings

Established an upper tail bound for solutions of stochastic difference equations.

The bound matches known lower bounds in the non-negative case.

Improves theoretical understanding of perpetuities' tail behavior.

Abstract

We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the non--negative case our bound is similar to a lower bound obtained by Goldie and Gr\"ubel in 1996.