Perpetuities with thin tails

Abstract

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

Tail asymptotics for exponential functionals of Lévy processes

“…behavior of R can be found in Goldie and Gru¨bel [19]. Other recent results on this equation leading to Pareto tails can be found in Konstantinides and Mikosch [25] and references therein.…”

Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

“…which proves (18). Inequality (19) has been obtained as follows: f does not increase, the sequence |Π k (ω)|, k = 0, 1, .…”

“…(b) if P{|M | ≤ 1} = 1 and Ee |Q| < ∞ for some > 0, then Ee ρ|Z∞| < ∞ for 0 ≤ ρ < sup{θ : Ee θ|Q| |M | < 1} (this fact follows from Theorem 2.1 in [18]; this work implicitly contains some other results related to moments).…”

Importance sampling of compounding processes

“…An obvious area is mathematical finance, but there are also applications in physics, communication networks, sorting algorithms, number theory, and more. For examples, see Vervaat (1979), Embrechts and Goldie (1994), Goldie and Grübel (1996). In particular, perpetuities naturally appear in time series analysis.…”

“…There is a substantial amount of literature on this topic; a non-exhaustive list is Blanchet and Glynn (2005), Goldie (1991), Goldie and Grübel (1996), Konstantinides and Mikosch (2005) and Maulik and Zwart (2006). Depending on the distribution of X 1 and B 1 , there can be many different types of tail behavior, ranging from extremely heavy (for example 1/ log x), to extremely light even inducing bounding support.…”