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On a stochastic difference equation and a representation of non–negative infinitely divisible random variables
Abstract
The present paper considers the stochastic difference equation Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f(x) = e −x/α.
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Cites
An introduction to probability theory and its applications
(1958) · Journal of the Franklin Institute · doi:10.1016/0016-0032(58)90688-4 · Cited by 29,269
An Introduction to Probability Theory and its Applications.
Kingman, J. F. C., & Feller, W. (1972) · Journal of the Royal Statistical Society. Series A (General) · doi:10.2307/2344620 · Cited by 4,938
A First Course in Stochastic Processes
(1975) · Elsevier eBooks · doi:10.1016/c2009-1-28569-8 · Cited by 3,890
The Theory of Stochastic Processes
Cox, D. R., Miller, H. D., Weiss, G., & Shuler, K. (1966) · Physics Today · doi:10.1063/1.3047872 · Cited by 3,324
A Course in Probability Theory
(1974) · Elsevier eBooks · doi:10.1016/c2009-0-22097-6 · Cited by 2,668
Sums of independent random variables
(1988) · Fourier Analysis · doi:10.1017/cbo9781107049949.052 · Cited by 1,755
An Introduction to Probability Theory and Its Applications
Freedman, D. A., & Feller, W. (1967) · Econometrica · doi:10.2307/1905668 · Cited by 1,244
The Theory of Stochastic Processes III
Gihman, I. I., & Skorohod, A. V. (1979) · Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-1-4615-8065-2 · Cited by 1,184
Random difference equations and Renewal theory for products of random matrices
Kesten, H. (1973) · Acta Mathematica · doi:10.1007/bf02392040 · Cited by 989
A cluster process representation of a self-exciting process
Hawkes, A. G., & Oakes, D. (1974) · Journal of Applied Probability · doi:10.1017/s0021900200096273 · Cited by 757
Cited by
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
Pitman, J., & Yor, M. (1997) · The Annals of Probability · doi:10.1214/aop/1024404422 · Cited by 1,202
Power Laws in Economics and Finance
Gabaix, X. (2009) · Annual Review of Economics · doi:10.1146/annurev.economics.050708.142940 · Cited by 907
Stationarity of Garch processes and of some nonnegative time series
Bougerol, P., & Picard, N. (1992) · Journal of Econometrics · doi:10.1016/0304-4076(92)90067-2 · Cited by 567
(2017) · Stochastic Processes · doi:10.1201/9781315381619-26 · Cited by 434
Metadata
- Authors
- Vervaat, W.
- Venue
- Advances in Applied Probability
- Publisher
- Cambridge University Press
- Published
- 1979-12-01
- Volume
- 11
- Issue
- 04
- Pages
- 750-783
- ISSN
- 0001-8678 (ISSN-L)
- OpenAlex
- W2334302903
- Cited by
- 462
- References
- 19
- FWCI
- 4.09
- Type
- journal article
- Language
- en
- Open access
- closed
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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