The Annals of Probability

Large deviations for solutions to stochastic recurrence equations under Kesten’s condition

D. Buraczewski, E. Damek, T. Mikosch, and J. Zienkiewicz

Full-text: Open access


In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten’s condition [Acta Math. 131 (1973) 207–248] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise sequence of the equations can have light tails. The results of the paper are analogs to those obtained by A. V. Nagaev [Theory Probab. Appl. 14 (1969) 51–64; 193–208] and S. V. Nagaev [Ann. Probab. 7 (1979) 745–789] in the case of partial sums of i.i.d. random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies in the sequence. We apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model.

Article information

Ann. Probab., Volume 41, Number 4 (2013), 2755-2790.

First available in Project Euclid: 3 July 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 91B30: Risk theory, insurance 60G70: Extreme value theory; extremal processes

Stochastic recurrence equation large deviations ruin probability


Buraczewski, D.; Damek, E.; Mikosch, T.; Zienkiewicz, J. Large deviations for solutions to stochastic recurrence equations under Kesten’s condition. Ann. Probab. 41 (2013), no. 4, 2755--2790. doi:10.1214/12-AOP782.

Export citation


  • [1] Bartkiewicz, K., Jakubowski, A., Mikosch, T. and Wintenberger, O. (2011). Stable limits for sums of dependent infinite variance random variables. Probab. Theory Related Fields 150 337–372.
  • [2] Cline, D. B. H. and Hsing, T. (1998). Large deviation probabilities for sums of random variables with heavy or subexponential tails. Technical report, Texas A&M Univ, College Station, TX.
  • [3] 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.
  • [4] de Haan, L., Resnick, S. I., Rootzén, H. and de Vries, C. G. (1989). Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Process. Appl. 32 213–224.
  • [5] Denisov, D., Dieker, A. B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: The big-jump domain. Ann. Probab. 36 1946–1991.
  • [6] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [7] Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55–72.
  • [8] Gantert, N. (2000). A note on logarithmic tail asymptotics and mixing. Statist. Probab. Lett. 49 113–118.
  • [9] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44 423–453.
  • [10] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.
  • [11] Hult, H., Lindskog, F., Mikosch, T. and Samorodnitsky, G. (2005). Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Probab. 15 2651–2680.
  • [12] Jakubowski, A. (1993). Minimal conditions in $p$-stable limit theorems. Stochastic Process. Appl. 44 291–327.
  • [13] Jakubowski, A. (1997). Minimal conditions in $p$-stable limit theorems. II. Stochastic Process. Appl. 68 1–20.
  • [14] Jessen, A. H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80 171–192.
  • [15] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
  • [16] Konstantinides, D. G. and Mikosch, T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33 1992–2035.
  • [17] 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.
  • [18] Mikosch, T. and Wintenberger, O. (2011). Precise large deviations for dependent regularly varying sequences. Unpublished manuscript.
  • [19] Nagaev, A. V. (1969). Integral limit theorems for large deviations when Cramér’s condition is not fulfilled I, II. Theory Probab. Appl. 14 51–64; 193–208.
  • [20] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745–789.
  • [21] Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. Oxford Univ. Press, New York.