## The Annals of Probability

### Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

#### Abstract

In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence $Y_{n}:=B_{1}+A_{1}B_{2}+\cdots+(A_{1}\cdots A_{n-1})B_{n}$, where $(A_{i},B_{i})\subset(0,\infty)\times\mathbb{R}$. Estimates for the stationary tail distribution of $\{Y_{n}\}$ have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207–248] and Goldie [Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that if $M:=\sup_{n}Y_{n}$, then $\mathbb{P}\{M>u\}\sim\mathcal{C}_{M}u^{-\xi}$ as $u\to\infty$. While much attention has been focused on extending such estimates to more general settings, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the normalized first passage time $T_{u}:=(\log u)^{-1}\inf\{n:Y_{n}>u\}$. We begin by showing that, conditional on $\{T_{u}<\infty\}$, $T_{u}\to\rho$ as $u\to\infty$ for a certain positive constant $\rho$. We then provide a conditional central limit theorem for $\{T_{u}\}$, and study $\mathbb{P}\{T_{u}\in G\}$ as $u\to\infty$ for sets $G\subset[0,\infty)$. If $G\subset[0,\rho)$, then we show that $\mathbb{P}\{T_{u}\in G\}u^{I(G)}\to C(G)$ as $u\to\infty$ for a certain large deviation rate function $I$ and constant $C(G)$. On the other hand, if $G\subset(\rho,\infty)$, then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac [In Random Matrices and Their Applications (Brunswick, Maine, 1984) (1986) 263–273 Amer. Math. Soc.], namely to the reflected process $M_{n}^{\ast}:=\max\{A_{n}M_{n-1}^{\ast}+B_{n},0\}$, $n\in\mathbb{Z}_{+}$. Using Siegmund duality, we relate the first passage times of $\{Y_{n}\}$ to the finite-time exceedance probabilities of $\{M_{n}^{\ast}\}$, yielding a new result concerning the convergence of $\{M_{n}^{\ast}\}$ to its stationary distribution.

#### Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3688-3739.

Dates
Revised: August 2015
First available in Project Euclid: 14 November 2016

https://projecteuclid.org/euclid.aop/1479114261

Digital Object Identifier
doi:10.1214/15-AOP1059

Mathematical Reviews number (MathSciNet)
MR3572322

Zentralblatt MATH identifier
1362.60023

#### Citation

Buraczewski, Dariusz; Collamore, Jeffrey F.; Damek, Ewa; Zienkiewicz, Jacek. Large deviation estimates for exceedance times of perpetuity sequences and their dual processes. Ann. Probab. 44 (2016), no. 6, 3688--3739. doi:10.1214/15-AOP1059. https://projecteuclid.org/euclid.aop/1479114261

#### References

• Alsmeyer, G. (2003). On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results. J. Theoret. Probab. 16 217–247.
• Alsmeyer, G. and Iksanov, A. (2009). A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron. J. Probab. 14 289–312.
• Alsmeyer, G. and Mentemeier, S. (2012). Tail behaviour of stationary solutions of random difference equations: The case of regular matrices. J. Difference Equ. Appl. 18 1305–1332.
• Arfwedson, G. (1955). Research in collective risk theory. Part II. Skand. Aktuarietidskr. 38 53–100.
• Asmussen, S. (2000). Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability 2. World Scientific, River Edge, NJ.
• Asmussen, S. and Sigman, K. (1996). Montone stochastic recursions and their duals. Probab. Engrg. Inform. Sci. 10 1–20.
• Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493–501.
• Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
• Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307–327.
• Brofferio, S. and Buraczewski, D. (2015). On unbounded invariant measures of stochastic dynamical systems. Ann. Probab. 43 1456–1492.
• Buraczewski, D. (2009). On tails of fixed points of the smoothing transform in the boundary case. Stochastic Process. Appl. 119 3955–3961.
• Buraczewski, D., Damek, E. and Zienkiewicz, J. (2015). Precise tail asymptotics of fixed points of the smoothing transform with general weights. Bernoulli 21 489–504.
• Buraczewski, D., Damek, E., Guivarc’h, Y., Hulanicki, A. and Urban, R. (2009). Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Related Fields 145 385–420.
• Buraczewski, D., Damek, E., Mikosch, T. and Zienkiewicz, J. (2013). Large deviations for solutions to stochastic recurrence equations under Kesten’s condition. Ann. Probab. 41 2755–2790.
• Buraczewski, D., Damek, E., Guivarc’h, Y. and Mentemeier, S. (2014). On multidimensional Mandelbrot cascades. J. Difference Equ. Appl. 20 1523–1567.
• Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes (O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, eds.) 41–55. Birkhäuser, Boston, MA.
• Collamore, J. F. (1998). First passage times of general sequences of random vectors: A large deviations approach. Stochastic Process. Appl. 78 97–130.
• Collamore, J. F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Probab. 19 1404–1458.
• Collamore, J. F. and Vidyashankar, A. N. (2013a). Large deviation tail estimates and related limit laws for stochastic fixed point equations. In Random Matrices and Iterated Random Functions. Springer Proc. Math. Stat. 53 91–117. Springer, Heidelberg.
• Collamore, J. F. and Vidyashankar, A. N. (2013b). Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stochastic Process. Appl. 123 3378–3429.
• Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston, MA.
• Ellis, R. S. (1984). Large deviations for a general class of random vectors. Ann. Probab. 12 1–12.
• 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. Probab. Theory Related Fields 144 581–613.
• Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3 349–375.
• Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.
• Guivarc’h, Y. (1990). Sur une extension de la notion de loi semi-stable. Ann. Inst. Henri Poincaré Probab. Stat. 26 261–285.
• Guivarc’h, Y. and Le Page, É. (2013a). Homogeneity at infinity of stationary solutions of multivariate affine stochastic recursions. In Random Matrices and Iterated Random Functions. Springer Proc. Math. Stat. 53 119–135. Springer, Heidelberg.
• Guivarc’h, Y. and Le Page, É. (2013b). On the homogeneity at infinity of the stationary probability for an affine random walk. Available at https://hal.archives-ouvertes.fr/hal-00868944.
• Iglehart, D. L. (1972). Extreme values in the $GI/G/1$ queue. Ann. Math. Stat. 43 627–635.
• Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
• Klüppelberg, C. and Kostadinova, R. (2008). Integrated insurance risk models with exponential Lévy investment. Insurance Math. Econom. 42 560–577.
• Klüppelberg, C. and Pergamenchtchikov, S. (2004). The tail of the stationary distribution of a random coefficient $\mathrm{AR}(q)$ model. Ann. Appl. Probab. 14 971–1005.
• Lalley, S. P. (1984). Limit theorems for first-passage times in linear and nonlinear renewal theory. Adv. in Appl. Probab. 16 766–803.
• Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications (Brunswick, Maine, 1984). Contemp. Math. 50 263–273. Amer. Math. Soc., Providence, RI.
• Liu, Q. (2000). On generalized multiplicative cascades. Stochastic Process. Appl. 86 263–286.
• Mikosch, T. (2003). Modeling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications, and the Environment (B. Finkenstädt and H. Rootzén, eds.) 185–286. Chapman & Hall, Boca Raton, FL.
• Mirek, M. (2011). Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Theory Related Fields 151 705–734.
• Nummelin, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 309–318.
• Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge Univ. Press, Cambridge.
• Nummelin, E. and Tuominen, P. (1982). Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stochastic Process. Appl. 12 187–202.
• Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stochastic Process. Appl. 92 265–285.
• Paulsen, J. (2002). On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann. Appl. Probab. 12 1247–1260.
• Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10 287–298.
• Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford Studies in Probability 4. Clarendon Press, New York.
• Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
• Roitershtein, A. (2007). One-dimensional linear recursions with Markov-dependent coefficients. Ann. Appl. Probab. 17 572–608.
• Siegmund, D. (1975). The time until ruin in collective risk theory. Mitt. Verein. Schweiz. Versicherungsmath. 75 157–166.
• Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Probab. 4 914–924.
• Varadhan, S. R. S. (1984). Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, PA.
• Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.
• von Bahr, B. (1974). Ruin probabilities expressed in terms of ladder height distributions. Scand. Actuar. J. 190–204.