The Annals of Probability

Stability of perpetuities

Charles M. Goldie and Ross A. Maller

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For a series of randomly discounted terms we give an integral criterion to distinguishbetween almost-sure absolute convergence and divergence in probability to $\infty$, 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.

Article information

Ann. Probab., Volume 28, Number 3 (2000), 1195-1218.

First available in Project Euclid: 18 April 2002

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Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80]

Distributional fixed point perpetuity random affine map randomly discounted series


Goldie, Charles M.; Maller, Ross A. Stability of perpetuities. Ann. Probab. 28 (2000), no. 3, 1195--1218. doi:10.1214/aop/1019160331.

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  • Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714-1730.
  • Brandt, A. (1986). The stochastic equation Yn+1 = AnYn + Bn withstationary coefficients. Adv. in Appl. Probab. 18 211-220.
  • Chow, Y.-S. and Zhang, C.-H. (1986). A note on Feller's strong law of large numbers. Ann. Probab. 14 1088-1094.
  • Embrechts, P. and Goldie, C. M. (1994). Perpetuities and random equations. In Asymptotic Statistics: Proceedings of the Fifth Prague Symposium (P. Mandl and M. Hu skov´a, eds.) 75-86. Physica, Heidelberg.
  • Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185 371-381.
  • Glasserman, P. and Yao, D. D. (1995). Stochastic vector difference equations with stationary coefficients. J. Appl. Probab. 32 851-866.
  • Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166.
  • Goldie, C. M. and Gr ¨ubel, R. (1996). Perpetuities withthin tails. Adv. in Appl. Probab. 28 463-480.
  • Grincevi cius, A. K. (1974). On the continuity of the distribution of a sum of dependent variables connected withindependent walks on lines. Teor. Veroyatnost. i Primen. 19 163-168. [Translation in Theory Probab. Appl. 19 (1974) 163-168.]
  • Grincevi cius, A. K. (1980). Products of random affine transformations. Litovsk. Mat. Sbornik 20 49-53. (Translation in Lithuanian Math. J. 20 279-282.)
  • Grincevi cius, A. K. (1981). A random difference equation. Litovsk. Mat. Sbornik 21 57-64. [Translation in Lithuanian Math. J. 21 (1981) 302-306.]
  • Gr ¨ubel, R. (1998). Hoare's selection algorithm: a Markov chain approach. J. Appl. Probab. 35 36-45.
  • Gr ¨ubel, R. and R ¨osler, U. (1996). Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Probab. 28 252-269.
  • Kellerer, H. G. (1992). Ergodic behaviour of affine recursions I: criteria for recurrence and transience; II: invariant measures and ergodic theorems; III: positive recurrence and null recurrence. Reports, Math. Inst. Univ. M ¨unchen, Theresienstrasse 39, D-8000 M ¨unchen, Germany.
  • Kesten, H. and Maller, R. A. (1996). Two renewal theorems for general random walks tending to infinity. Probab. Theory Related Fields 106 1-38.
  • Klass, M. J. and Wittman, R. (1993). Which i.i.d. sums are recurrently dominated by their maximal terms J. Theoret. Probab. 6 195-207.
  • Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications 263-273. Amer. Math. Soc., Providence, RI.
  • Lo eve, M. (1977). Probability Theory 1, 4thed. Springer, New York.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Univ. Press.
  • Pruitt, W. E. (1981). General one-sided laws of the iterated logarithm. Ann. Probab. 9 1-48.
  • Spitzer, F. (1976). Principles of Random Walk, 2nd ed. Springer, New York.
  • Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750-783.
  • Zakusilo, O. K. (1975). On classes of limit distributions in a summation scheme. Teor. Veroyatnost. i Mat. Statist. 12 44-48. (Translation in Theor. Probab. and Math. Statist. 12 44-48.)