# Tail behaviour of stationary solutions of random difference equations: the case of regular matrices

@article{Alsmeyer2012TailBO,
title={Tail behaviour of stationary solutions of random difference equations: the case of regular matrices},
author={Gerold Alsmeyer and Sebastian Mentemeier},
journal={Journal of Difference Equations and Applications},
year={2012},
volume={18},
pages={1305 - 1332}
}
• Published 9 September 2010
• Mathematics
• Journal of Difference Equations and Applications
Given a sequence of i.i.d. random variables with generic copy such that M is a regular matrix and Q takes values in , we consider the random difference equation Under suitable assumptions stated below, this equation has a unique stationary solution R such that for some and some finite positive and continuous function K on , holds true. A rather long proof of this result, originally stated by Kesten [Acta Math. 131 (1973), pp. 207–248] at the end of his famous article, was given by Le Page [S…

### On Multivariate Stochastic Fixed Point Equations: The Smoothing Transform and Random Difference Equations

The thesis at hand is concerned with the study of random vectors Y ∈ R d , satisfying multivariate stochastic fixed point equations Y d = N i=1 T i Y i + Q (d =:same distribution). Here N ≥ 1 fixed,

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

• Mathematics
• 2014
In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence Yn:=B1+A1B2+⋯+(A1⋯An−1)BnYn:=B1+A1B2+⋯+(A1⋯An−1)Bn,

### CHARACTERIZATION OF THE TAIL BEHAVIOR OF A CLASS OF BEKK PROCESSES: A STOCHASTIC RECURRENCE EQUATION APPROACH

• Mathematics
Econometric Theory
• 2021
We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary

### Quasistochastic matrices and Markov renewal theory

• G. Alsmeyer
• Mathematics
Journal of Applied Probability
• 2014
Main results include Markov renewal-type theorems and a Stone-type decomposition under an absolute continuity condition and three applications are given.

### Precise Large Deviation Results for Products of Random Matrices

• Mathematics
• 2014
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

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

• Mathematics, Computer Science
• 2016
This work characterize the distribution of the first passage time of the matrix recursive sequence V_n = M_n V_{n-1} + Q_n, showing that this distribution converges to the stationary law of the exponentially-shifted Markov random walk and describes the large exceedance paths via two conditioned limit laws.

### Random difference equations with subexponential innovations

• Mathematics
• 2016
We consider the random difference equations S =d (X + S)Y and T =dX + TY, where =d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side

### On the Kesten–Goldie constant

• Mathematics
• 2016
We consider the stochastic difference equation on where is an i.i.d. sequence of random variables and is an initial distribution. Under mild contractivity hypotheses the sequence converges in law to

### On the rate of convergence in the Kesten renewal theorem

• Mathematics
• 2015
We consider the stochastic recursionXn+1 = Mn+1Xn +Qn+1 on R d , where (Mn;Qn) are i.i.d. random variables such that Qn are translations, Mn are similarities of the Euclidean space R d . Under some

## References

SHOWING 1-10 OF 21 REFERENCES

### Random difference equations and Renewal theory for products of random matrices

where Mn and Qn are random d • d matrices respectively d-vectors and Yn also is a d-vector. Throughout we take the sequence of pairs (Mn, Q~), n >/1, independently and identically distributed. The

### Heavy tail properties of stationary solutions of multidimensional stochastic recursions

We consider the following recurrence relation with random i.i.d. coefficients $(a_n,b_n)$: $$x_{n+1}=a_{n+1} x_n+b_{n+1}$$ where $a_n\in GL(d,\mathbb{R}),b_n\in \mathbb{R}^d$. Under natural

### IMPLICIT RENEWAL THEORY AND TAILS OF SOLUTIONS OF RANDOM EQUATIONS

in which ,u is a known probability measure and g a known function. By "implicit renewal theory" is meant a variant in which g is not known and indeed is an integral involving f itself.

### Tail-homogeneity of stationary measures for some multidimensional stochastic recursions

• Mathematics
• 2008
We consider a stochastic recursion Xn+1 = Mn+1Xn + Qn+1, ($${n\in \mathbb {N}}$$), where (Qn, Mn) are i.i.d. random variables such that Qn are translations, Mn are similarities of the Euclidean space

### Products of Random Matrices with Applications to Schrödinger Operators

• Mathematics
• 1985
A: "Limit Theorems for Products of Random Matrices".- I - The Upper Lyapunov Exponent.- 1. Notation.- 2. The upper Lyapunov exponent.- 3. Cocycles.- 4. The theorem of Furstenberg and Kesten.- 5.

### Iterated Random Functions

• Mathematics
SIAM Rev.
• 1999
Survey of iterated random functions offers a method for studying the steady state distribution of a Markov chain, and presents useful bounds on rates of convergence in a variety of examples.

### The Markov Renewal Theorem and Related Results

We give a new probabilistic proof of the Markov renewal theorem for Markov random walks with positive drift and Harris recurrent driving chain. It forms an alternative to the one recently given in

### The tail of the stationary distribution of a random coefficient AR(q) model

• Mathematics
• 2004
We investigate a stationary random cofficient autoregressive process. Using renewal type arguments tailor-made for such processes we show that the stationary distribution has a power-law tail. When