Appendix to “Approximating Perpetuities”

Appendix to “Approximating perpetuities”

An algorithm for perfect simulation from the unique solution of the distributional fixed point equation Y =d UY +U(1−U) is constructed, where Y and U are independent and U is uniformly distributed on [0, 1]. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma ...

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...

Perpetuities with Thin Tails

Stability of Perpetuities

For a series of randomly discounted terms we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to 1, these being the only possible forms of asymptotic behaviour. This solves the existence problem for a one-dimensional perpetuity that remains from a 1979 study by Vervaat, and yields a complete characterization of the existence of dis...

Stabilität von Perpetuities