The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of large numbers for the norm of a vector on which the matrices act. We prove corresponding precise large deviation results, generalizing the Bahadur-Rao theorem to this situation. Therefore, we obtain a third-order Edgeworth expansion for the cumulative distribution function of the vector norm. This result in turn relies on an application of the Nagaev-Guivarch method. Our result is then used to study matrix recursions, arising e.g. in financial time series, and to provide precise large deviation estimates there.

Precise Large Deviation Results for Products of Random Matrices

Moments of the stationary distribution of subcritical multitype Galton–Watson processes with immigration

We determine the large exceedance probabilities and large exceedance paths for the matrix recursive sequence $V_n = M_n V_{n-1} + Q_n, \: n=1,2,\ldots,$ where $\{M_n\}$ is an i.i.d. sequence of $d \times d$ random matrices and $\{ Q_n\}$ is an i.i.d. sequence of random vectors, both with nonnegative entries. Early work on this problem dates to Kesten's (1973) seminal paper, motivated by an application to multi-type branching processes. Other applications arise in financial time series modeling (connected to the study of the GARCH($p,q$) processes) and in physics, and this recursive sequence has also been the focus of extensive work in the recent probability literature. In this work, we characterize the distribution of the first passage time $T_u^A := \inf \{n: V_n \in u A \}$, where $A$ is a subset of the nonnegative quadrant in ${\mathbb R}^d$, showing that $T_u^A/u^\alpha$ converges to an exponential law. In the process, we also revisit and refine Kesten's classical estimate, showing that if $V$ has the stationary distribution of $\{ V_n \}$, then ${\mathbb P} \left( V \in uA \right) \sim C_A u^{-\alpha}$ as $u \to \infty$, providing, most importantly, a new characterization of the constant $C_A$. Finally, we describe the large exceedance paths via two conditioned limit laws. In the first, we show that conditioned on a large exceedance, the process $\{ V_n\}$ follows an exponentially-shifted Markov random walk, which we identify, thereby generalizing results for classical random walk to matrix recursive sequences. In the second, we characterize the empirical distribution of $\{ \log |V_n| - \log |V_{n-1}| \}$ prior to a large exceedance, showing that this distribution converges to the stationary law of the exponentially-shifted Markov random walk.

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

The results about branching structures within random walks will be surveyed at first. Then as an example of application, the stationary distribution of a birth-and-death process with bounded jumps will be calculated explicitly.

Branching Structures Within Random Walks and Their Applications

The degree sequence $(N_{i,j}(k),1\leq i,j\leq d,k\geq 0)$ of a multitype forest with $d$ types, is the number of individuals type $i$, having $k$ children type $j$. We construct a multitype forest sampled uniformly from all multitype forest with a given degree sequence (MFGDS). For this, we use an extension of the Ballot Theorem by (Chaumont and Liu, 2016), and generalize the Vervaat transform (Vervaat, 1979) to multidimensional discrete exchangeable increment processes. We prove that MFGDS are extensions of multitype Galton-Watson (MGW) forests, since mixing the laws of the former, one obtains MGW forests with fixed sizes by type (CMGW). We also obtain the law of the total population by types in a MGW forest, generalizing Otter-Dwass formula (Otter 1949, Dwass 1969). We apply this to obtain enumerations of plane, labeled and binary multitype forests having fixed roots and individuals by types. We give an algorithm to simulate certain CMGW forests, generalizing the unitype case of (Devroye, 2012).

On Multitype Random Forests with a Given Degree Sequence, the Total Population of Branching Forests and Enumerations of Multitype Forests

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t−κ as t → ∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see “Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145–168 (1975)”).

A note on multitype branching process with bounded immigration in random environment

In this paper, we form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the first passage time of corresponding random walks. Especially, for recurrent random walk which could stay at its position with probability 0 < r < 1, we show that the tail probability of the first passage time decays as $\frac{2} {{\sqrt {\pi (1 - r)} }}\frac{1} {{\sqrt n }} $ when n → ∞.

Hua Ming Wang

Papers

A note on multitype branching process with bounded immigration in random environment

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice