Stability of perpetuities

Charles M. Goldie; Ross A. Maller

doi:10.1214/aop/1019160331

Stability of perpetuities

Charles M. Goldie^*, Ross A. Maller

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

108 Citations (Scopus)

Abstract

For a series of randomly discounted terms we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to ∞, these being the only possible forms of asymptotic behavior. 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 distributional fixed points of a random affine map in dimension one.

Original language	English
Pages (from-to)	1195-1218
Number of pages	24
Journal	Annals of Probability
Volume	28
Issue number	3
DOIs	https://doi.org/10.1214/aop/1019160331
Publication status	Published - Jul 2000
Externally published	Yes

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10.1214/aop/1019160331

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title = "Stability of perpetuities",

abstract = "For a series of randomly discounted terms we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to ∞, these being the only possible forms of asymptotic behavior. 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 distributional fixed points of a random affine map in dimension one.",

keywords = "Distributional fixed point, Perpetuity, Random affine map, Randomly discounted series",

author = "Goldie, {Charles M.} and Maller, {Ross A.}",

year = "2000",

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doi = "10.1214/aop/1019160331",

language = "English",

volume = "28",

pages = "1195--1218",

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T1 - Stability of perpetuities

AU - Goldie, Charles M.

AU - Maller, Ross A.

PY - 2000/7

Y1 - 2000/7

N2 - For a series of randomly discounted terms we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to ∞, these being the only possible forms of asymptotic behavior. 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 distributional fixed points of a random affine map in dimension one.

AB - For a series of randomly discounted terms we give an integral criterion to distinguish between almost-sure absolute convergence and divergence in probability to ∞, these being the only possible forms of asymptotic behavior. 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 distributional fixed points of a random affine map in dimension one.

KW - Distributional fixed point

KW - Perpetuity

KW - Random affine map

KW - Randomly discounted series

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U2 - 10.1214/aop/1019160331

DO - 10.1214/aop/1019160331

M3 - Article

SN - 0091-1798

VL - 28

SP - 1195

EP - 1218

JO - Annals of Probability

JF - Annals of Probability

IS - 3

ER -

Stability of perpetuities

Abstract

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