## The Annals of Probability

### Large excursions and conditioned laws for recursive sequences generated by random matrices

#### Abstract

We study the large exceedance probabilities and large exceedance paths of the recursive sequence $V_{n}=M_{n}V_{n-1}+Q_{n}$, where $\{(M_{n},Q_{n})\}$ is an i.i.d. sequence, and $M_{1}$ is a $d\times d$ random matrix and $Q_{1}$ is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for $\{V_{n}\}$ and a Cramér-type condition for $\{M_{n}\}$. Under these assumptions, we characterize the distribution of the first passage time $T_{u}^{A}:=\inf\{n:V_{n}\in uA\}$, where $A$ is a general subset of $\mathbb{R}^{d}$, exhibiting that $T_{u}^{A}/u^{\alpha}$ converges to an exponential law for a certain $\alpha>0$. In the process, we revisit and refine classical estimates for $\mathbb{P}(V\in uA)$, where $V$ possesses the stationary law of $\{V_{n}\}$. Namely, for $A\subset\mathbb{R}^{d}$, we show that $\mathbb{P}(V\in uA)\sim C_{A}u^{-\alpha}$ as $u\to\infty$, providing, most importantly, a new characterization of the constant $C_{A}$. As a simple consequence of these estimates, we also obtain an expression for the extremal index of $\{|V_{n}|\}$. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that $\{V_{n}\}$ follows an exponentially-shifted Markov random walk, which we identify. We thereby generalize results from the theory of classical random walk to multivariate recursive sequences.

#### Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2064-2120.

Dates
Revised: July 2017
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.aop/1528876822

Digital Object Identifier
doi:10.1214/17-AOP1221

Mathematical Reviews number (MathSciNet)
MR3813986

Zentralblatt MATH identifier
06919019

#### Citation

Collamore, Jeffrey F.; Mentemeier, Sebastian. Large excursions and conditioned laws for recursive sequences generated by random matrices. Ann. Probab. 46 (2018), no. 4, 2064--2120. doi:10.1214/17-AOP1221. https://projecteuclid.org/euclid.aop/1528876822

#### References

• [1] Alsmeyer, G. (1991). Erneuerungstheorie: Analyse stochastischer Regenerationsschemata. B. G. Teubner, Stuttgart.
• [2] Alsmeyer, G. (1997). The Markov renewal theorem and related results. Markov Process. Related Fields 3 103–127.
• [3] Alsmeyer, G. (2003). On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results. J. Theoret. Probab. 16 217–247.
• [4] Alsmeyer, G. (2016). On the stationary tail index of iterated random Lipschitz functions. Stochastic Process. Appl. 126 209–233.
• [5] Asmussen, S. (1982). Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the $GI/G/1$ queue. Adv. in Appl. Probab. 14 143–170.
• [6] Basrak, B., Davis, R. A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908–920.
• [7] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Probab. 22 2152–2167.
• [8] Boman, J. and Lindskog, F. (2009). Support theorems for the Radon transform and Cramér–Wold theorems. J. Theoret. Probab. 22 683–710.
• [9] Buraczewski, D., Collamore, J. F., Damek, E. and Zienkiewicz, J. (2016). Large deviation estimates for exceedance times of perpetuity sequences and their dual processes. Ann. Probab. 44 3688–3739.
• [10] Buraczewski, D., Damek, E., Guivarc’h, Y. and Mentemeier, S. (2014). On multidimensional Mandelbrot cascades. J. Difference Equ. Appl. 20 1523–1567.
• [11] Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power Law Tails: The Equation $X=AX+B$. Springer, Berlin.
• [12] Buraczewski, D., Damek, E. and Zienkiewicz, J. (2016). On the Kesten–Goldie constant. J. Difference Equ. Appl. 22 1646–1662.
• [13] Buraczewski, D. and Mentemeier, S. (2016). Precise large deviation results for products of random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 52 1474–1513.
• [14] Collamore, J. F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Probab. 19 1404–1458.
• [15] Collamore, J. F., Diao, G. and Vidyashankar, A. N. (2014). Rare event simulation for processes generated via stochastic fixed point equations. Ann. Appl. Probab. 24 2143–2175.
• [16] Collamore, J. F. and Mentemeier, S. (2016). Large excursions and conditioned laws for recursive sequences generated by random matrices. Preprint. Available at arXiv:1608.05175.
• [17] Collamore, J. F. and Vidyashankar, A. N. (2013). Large deviation tail estimates and related limit laws for stochastic fixed point equations. In Random Matrices and Iterated Random Functions (G. Alsmeyer and M. Löwe, eds.) 91–117. Springer, Berlin.
• [18] Collamore, J. F. and Vidyashankar, A. N. (2013). Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stochastic Process. Appl. 123 3378–3429.
• [19] 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.
• [20] Dembo, A., Karlin, S. and Zeitouni, O. (1994). Large exceedances for multidimensional Lévy processes. Ann. Appl. Probab. 4 432–447.
• [21] 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.
• [22] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York.
• [23] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.
• [24] Guivarc’h, Y. (1990). Sur une extension de la notion de loi semi-stable. Ann. Inst. Henri Poincaré Probab. Stat. 26 261–285.
• [25] Guivarc’h, Y. and Le Page, É. (2016). Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions. Ann. Inst. Henri Poincaré Probab. Stat. 52 503–574.
• [26] Hennion, H. (1997). Limit theorems for products of positive random matrices. Ann. Probab. 25 1545–1587.
• [27] Iglehart, D. L. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Stat. 43 627–635.
• [28] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
• [29] Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2 355–386.
• [30] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compos. Math. 30 145–168.
• [31] Leadbetter, M. R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Probab. 16 431–478.
• [32] 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.
• [33] Liu, Q. (2000). On generalized multiplicative cascades. Stochastic Process. Appl. 86 263–286.
• [34] Melfi, V. F. (1992). Nonlinear Markov renewal theory with statistical applications. Ann. Probab. 20 753–771.
• [35] Melfi, V. F. (1994). Nonlinear renewal theory for Markov random walks. Stochastic Process. Appl. 54 71–93.
• [36] Mentemeier, S. (2013). On Multivariate Stochastic Fixed Point Equations: The Smoothing Transform and Random Difference Equations. Ph.D. thesis, Westfälische Wilhelms-Universität Münster.
• [37] Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.
• [38] 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.
• [39] Mirek, M. (2011). Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Theory Related Fields 151 705–734.
• [40] Nummelin, E. (1984). General Irreducible Markov Chains and Non-negative Operators. Cambridge Univ. Press, Cambridge.
• [41] Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Probab. 4 529–548.
• [42] Perfekt, R. (1997). Extreme value theory for a class of Markov chains with values in ${\mathbb{R}}^{d}$. Adv. in Appl. Probab. 29 138–164.
• [43] Resnick, S. (2004). On the foundations of multivariate heavy-tail analysis. J. Appl. Probab. 41A 191–212.
• [44] Roitershtein, A. (2007). One-dimensional linear recursions with Markov-dependent coefficients. Ann. Appl. Probab. 17 572–608.
• [45] Shurenkov, V. M. (1984). On Markov renewal theory. Theory Probab. Appl. 29 247–265.
• [46] Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
• [47] Solomon, F. (1972). Random walks in random environment. Ph.D. dissertation, Cornell Univ., Ithaca, NY.
• [48] Solomon, F. (1975). Random walks in random environment. Ann. Probab. 3 1–31.
• [49] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
• [50] Wang, H. M. (2013). A note on multitype branching process with bounded immigration in random environment. Acta Math. Sin. (Engl. Ser.) 29 1095–1110.