# Simulating Perpetuities

*Luc Devroye* ()

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Luc Devroye: McGill University

*Methodology and Computing in Applied Probability*, 2001, vol. 3, issue 1, 97-115

**Abstract:**
Abstract A perpetuity is a random variable that can be represented as $$1 + W_1 + W_1 W_2 + W_1 W_2 W_3 + \cdot \cdot \cdot ,$$ , 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 $$W_1 \underline{\underline {\mathcal{L}}} {\text{ }}U^{1/\beta } ,\beta > 0,U$$ uniform [0, 1], all the details of a novel rejection method are worked out. There exists an implementation of our algorithm that only uses uniform random numbers, additions, multiplications and comparisons.

**Keywords:** random variate generation; perpetuities; rejection method; simulation; monte carlo method; expected time analysis; probability inequalities; infinite divisiblity (search for similar items in EconPapers)

**Date:** 2001

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**DOI:** 10.1023/A:1011470225335

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