Journal of Applied Probability

Fractional moments of solutions to stochastic recurrence equations

Thomas Mikosch, Gennady Samorodnitsky, and Laleh Tafakori

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In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt-1 + Bt, tZ, where ((At, Bt))tZ is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, pR. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

Article information

J. Appl. Probab., Volume 50, Number 4 (2013), 969-982.

First available in Project Euclid: 10 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K99: None of the above, but in this section

Moment stochastic recurrence equation GARCH Erlang distribution numerical approximation


Mikosch, Thomas; Samorodnitsky, Gennady; Tafakori, Laleh. Fractional moments of solutions to stochastic recurrence equations. J. Appl. Probab. 50 (2013), no. 4, 969--982. doi:10.1239/jap/1389370094.

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  • Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On \ds al properties of perpetuities. J. Theoret. Prob. 22, 666–682.
  • Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Cambridge University Press.
  • Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.
  • Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.
  • Behme, A., Lindner, A. and Maller, R. (2011). Stationary solutions of the stochastic differential equation $dV_t=V_{t^-}dU_t+dL_t$ with Lévy noise. Stoch. Process. Appl. 121, 91–108.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.
  • Boxma, O., Kella, O. and Perry, D. (2011). On some tractable growth-collapse processes with renewal collapse epochs. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A), eds P. Glynn, T. Mikosch and T. Rolski, Applied Probability Trust, Sheffield, pp. 217–234.
  • Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.
  • Brockwell, P. J. and Lindner, A. (2009). Existence and uniqueness of stationary Lévy-driven CARMA processes. Stoch. Process. Appl. 119, 2660–2681.
  • Carmona, P., Petit, F. and Yor, M. (1997). On the \ds and \asy results for exponential \fct als of \levy processes. In Exponential Functionals and Principle Values Related to \BM, ed. M. Yor, Revista Matematica Iberoamericana, Madrid, pp. 73–130.
  • Collamore, J. F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Prob. 19, 1404–1458.
  • Diaconis, P. and Freedman, D. (1999). Iterated random \fct s. SIAM Rev. 41, 45–76.
  • Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.
  • Dufresne, D. (1996). On the stochastic equation $\mathcal{L}(X)=\mathcal{L}[B(X+C)]$ and a property of gamma distributions. Bernoulli 2, 287–291.
  • Dufresne, D. (1998). Algebraic properties of beta and gamma \ds s, and applications. Adv. Appl. Math. 20, 285–299.
  • Dufresne, D. (2010). $G$ distributions and the beta-gamma algebra. Electron. J. Prob. 15, 2163–2199.
  • Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. Appl. Prob. 34, 85–111.
  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.
  • Enriquez, N., Sabot, C. and Zindy, O. (2009). A probabilistic representation of constants in Kesten's renewal theorem. Prob. Theory Relat. Fields 144, 581–613.
  • Gjessing, H. K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144.
  • Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.
  • Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.
  • Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential \fct als. Ann. Appl. Prob. 14, 90–117.
  • Hirsch, F. and Yor, M. (2013). On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator. Bernoulli 19, 1350–1377.
  • Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.
  • Klebanov, L. B., Maniya, G. M. and Melamed, I. A. (1985). A problem of Zolotarev and analogs of infinitely divisible and stable \ds s in a scheme for summing a random number of \rv s. Theory Prob. Appl. 29, 791–794.
  • Kozubowski, T. J. (2000). Exponential mixture \rep of geometric stable \ds s. Ann. Inst. Statist. Math. 52, 231–238.
  • Löpker, A. H. and van Leeuwaarden, J. S. H. (2008). Transient moments of the TCP window size process. J. Appl. Prob. 45, 163–175.
  • Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential \fct als of \levy processes. Stoch. Process. Appl. 116, 156–177.
  • Pitman, J. and Yor, M. (2003). Infinitely divisible laws associated with hyperbolic \fct s. Canad. J. Math. 55, 292–330.
  • Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. John Wiley, Chichester.
  • Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.
  • Vervaat, W. (1979). On a stochastic difference equation and a \rep of nonnegative infinitely divisible \rv s. Adv. Appl. Prob. 11, 750–783.
  • Zolotarev, V. M. (1986). One-Dimensional Stable Distributions (Trans. Math. Monogr. 65). American Mathematical Society, Providence, RI.