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

TY - JOUR

T1 - Large excursions and conditioned laws for recursive sequences generated by random matrices

AU - Collamore, Jeffrey F.

AU - Mentemeier, Sebastian

N1 - To appear in The Annals of Probability

PY - 2018

Y1 - 2018

N2 - We study the large exceedance probabilities and large exceedance paths of the recursive sequence Vn =MnVn-1 + Qn, where ((Mn,Qn)) is an i.i.d. sequence, and M1 is a d × d random matrix and Q1 is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for (Vn) and a Cramér-type condition for (Mn). Under these assumptions, we characterize the distribution of the first passage time TuA := inf(n: Vn ∞ uA), where A is a general subset of ℝd, exhibiting that TuA/uα converges to an exponential law for a certain α > 0. In the process, we revisit and refine classical estimates for P(V ∞ uA), where V possesses the stationary law of (Vn). Namely, for A ⊂ ℝd, we show that P(V ∞ uA) ~ CAu-a as u→∞, providing, most importantly, a new characterization of the constant CA. As a simple consequence of these estimates, we also obtain an expression for the extremal index of (|Vn|). Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that (Vn) 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.

AB - We study the large exceedance probabilities and large exceedance paths of the recursive sequence Vn =MnVn-1 + Qn, where ((Mn,Qn)) is an i.i.d. sequence, and M1 is a d × d random matrix and Q1 is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for (Vn) and a Cramér-type condition for (Mn). Under these assumptions, we characterize the distribution of the first passage time TuA := inf(n: Vn ∞ uA), where A is a general subset of ℝd, exhibiting that TuA/uα converges to an exponential law for a certain α > 0. In the process, we revisit and refine classical estimates for P(V ∞ uA), where V possesses the stationary law of (Vn). Namely, for A ⊂ ℝd, we show that P(V ∞ uA) ~ CAu-a as u→∞, providing, most importantly, a new characterization of the constant CA. As a simple consequence of these estimates, we also obtain an expression for the extremal index of (|Vn|). Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that (Vn) 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.

U2 - 10.1214/17-aop1221

DO - 10.1214/17-aop1221

M3 - Journal article

SN - 0091-1798

VL - 46

SP - 2064-2120.

JO - Annals of Probability

JF - Annals of Probability

IS - 4

ER -