The Annals of Probability

Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations

Dimitrios G. Konstantinides and Thomas Mikosch

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In this paper we consider the stochastic recurrence equation Yt=AtYt−1+Bt for an i.i.d. sequence of pairs (At,Bt) of nonnegative random variables, where we assume that Bt is regularly varying with index κ>0 and EAtκ<1. We show that the stationary solution (Yt) to this equation has regularly varying finite-dimensional distributions with index κ. This implies that the partial sums Sn=Y1+⋯+Yn of this process are regularly varying. In particular, the relation P(Sn>x)∼c1nP(Y1>x) as x→∞ holds for some constant c1>0. For κ>1, we also study the large deviation probabilities P(SnESn>x), xxn, for some sequence xn→∞ whose growth depends on the heaviness of the tail of the distribution of Y1. We show that the relation P(SnESn>x)∼c2nP(Y1>x) holds uniformly for xxn and some constant c2>0. Then we apply the large deviation results to derive bounds for the ruin probability ψ(u)=P(sup n≥1((SnESn)−μn)>u) for any μ>0. We show that ψ(u)∼c3uP(Y1>u−1(κ−1)−1 for some constant c3>0. In contrast to the case of i.i.d. regularly varying Yt’s, when the above results hold with c1=c2=c3=1, the constants c1, c2 and c3 are different from 1.

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Ann. Probab., Volume 33, Number 5 (2005), 1992-2035.

First available in Project Euclid: 22 September 2005

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Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 91B30: Risk theory, insurance 60G70: Extreme value theory; extremal processes 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Stochastic recurrence equation large deviations regular variation ruin probability


Konstantinides, Dimitrios G.; Mikosch, Thomas. Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33 (2005), no. 5, 1992--2035. doi:10.1214/009117905000000350.

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  • Basrak, B., Davis, R. A. and Mikosch. T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95--116.
  • Baxendale, P. H. and Khasminskii, R. Z. (1998). Stability index for products of random transformations. Adv. in Appl. Probab. 30 968--988.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714--1730.
  • Boussama, F. (1998). Ergodicité, mélange et estimation dans le modelès GARCH. Ph.D. thesis, Univ. Paris 7.
  • Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323--331.
  • Cline, D. B. H. and Hsing, T. (1991). Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Texas A&M Univ. Preprint.
  • Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879--917.
  • Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049--2080.
  • Davis, R. A. and Mikosch, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelations. J. Appl. Probab. Special Volume: A Festschrift for David Vere-Jones 38A 93--104.
  • Davis, R. A. and Resnick, S. I. (1996). Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Probab. 6 1191--1210.
  • Doukhan, P. (1994). Mixing. Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
  • Dufresne, D. (1990). The distribution of a perpetuity, with application to risk theory. Scand. Actuar. J. 39--79.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
  • Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. in Appl. Probab. 28 463--480.
  • Grey, D. R. (1994). Regular variation in the tail behaviour of solutions to random difference equations. Ann. Appl. Probab. 4 169--183.
  • de Haan, L., Resnick, S. I., Rootzén, H. and de Vries, C. (1989). Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Process. Appl. 32 213--224.
  • Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters--Noordhoff, Groningen.
  • Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207--248.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Mikosch, T. (2003). Modeling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications, and the Environment (B. Finkenstädt and H. Rootén, eds.) 185--286. Chapman and Hall, Boca Raton.
  • Mikosch, T. and Nagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications to insurance. Extremes 1 81--110.
  • Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 1025--1064.
  • Mikosch, T. and Samorodnitsky, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Probab. 28 1814--1851.
  • Mikosch, T. and Straumann, D. (2005). Stable limits of martingale transforms with application to the estimation of GARCH parameters. Ann. Statist. To appear.
  • Mokkadem, A. (1990). Propriétés de mélange des processus autorégressifs polynomiaux. Ann. Inst. H. Poincaré Probab. Statist. 26 219--260.
  • Nagaev, A. V. (1969). Limit theorems for large deviations when Cramér's conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz.--Mat. Nauk 6 17--22. (In Russian.)
  • Nagaev, S. V. (1979). Large deviations of sums independent random variables. Ann. Probab. 7 745--789.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Resnick, S. I. and Willekens, E. (1991). Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statistics: Stochastic Models 7 511--525.
  • Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43--47.
  • Samorodnitsky, G. (2002). Long Range Dependence, Heavy Tails and Rare Events. MaPhySto Lecture Notes. Available at
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall, London.
  • Straumann, D. (2003). Estimation in conditonally heteroscedastic time series models. Ph.D. thesis, Institute of Mathematical Science, Univ. Copenhagen.
  • Straumann, D. and Mikosch, T. (2005). Quasi-maximum likelihood estimation in heteroscedastic time series: A stochastic recurrence equations approach. Ann. Statist. To appear.