On perpetuities with light tails

Perpetuities with Thin Tails

Extremes of Lévy Processes with Light Tails

Let X(t) t ≥ 0 , X(0) = 0, be a Lévy process with spectral Lévy measure ρ. Assuming that ρ((−∞, 0)) < ∞ and the right tail of ρ is light, we show that in the presence of Brownian component P „ sup 0≤t≤1 X(t) > u « ∼ P (X(1) > u) as u → ∞. In the absence of Brownian component these tails are not always comparable. An example of Lévy process of the type X(t) = B(t) + Z(t), where B(t) is a Brownia...

On distributional properties of perpetuities

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of p-moments, p > 0 as well as exponential moments. In particular, a formula fo...

From light tails to heavy tails through multiplier

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class L(γ ) or S(γ ) for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class L(0) or S(0) accordingly. Hence, the multiplier Y builds a bridge between ...

Simulating Perpetuities

A perpetuity is a random variable that can be represented as 1 + W 1 + W 1 W 2 + W 1 W 2 W 3 + , where the W i 's are i.i.d. random variables. We study exact random variate generation for perpetuities and discuss the expected complexity. For the Vervaat family, in which W 1 L = U 1== , > 0, U uniform 0; 1], all the details of a novel rejection method are worked out. There exists an implementati...