# On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case

@article{Brofferio2008OnTI,
title={On the invariant measure of the random difference equation \$X\_n=A\_n X\_\{n-1\}+ B\_n\$ in the critical case},
author={S. Brofferio and Dariusz Buraczewski and Ewa Damek},
journal={arXiv: Probability},
year={2008}
}
• Published 10 September 2008
• Mathematics
• arXiv: Probability
We consider the autoregressive model on $\R^d$ defined by the following stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d. random variables valued in $\R^d\times \R^+$. The critical case, when $\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure $\nu$ for the Markov chain $\{X_n \}$. In the present paper we prove that the weak limit of properly dilated measure $\nu$ exists and defines a…
On unbounded invariant measures of stochastic dynamical systems
• Mathematics
• 2013
We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=\Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi _n$ are i.i.d. random continuous transformations of
On the derivative martingale in a branching random walk
• Mathematics
• 2020
We work under the A\"{\i}d\'{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate
Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case
We consider the stochastic recursion $${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$$, where $${Q_n, X_n \in \mathbb{R}^d }$$, Mn are similarities of the Euclidean space $${ \mathbb{R}^d Recurrence of two-dimensional queueing processes, and random walk exit times from the quadrant • Mathematics The Annals of Applied Probability • 2021 Let X = (X_1, X_2) be a 2-dimensional random variable and X(n), n \in \mathbb{N} a sequence of i.i.d. copies of X. The associated random walk is S(n)= X(1) + \cdots +X(n). The corresponding Recurrence on affine Grassmannians • Mathematics Ergodic Theory and Dynamical Systems • 2019 We study the action of the affine group G of \mathbb{R}^{d} on the space X_{k,\,d} of k -dimensional affine subspaces. Given a compactly supported Zariski dense probability measure Anti-concentration for subgraph counts in random graphs • Mathematics The Annals of Probability • 2019 Fix a graph H and some p\in (0,1), and let X_H be the number of copies of H in a random graph G(n,p). Random variables of this form have been intensively studied since the foundational work Random iteration with place dependent probabilities • Mathematics • 2011 Markov chains arising from random iteration of functions S_{\theta}:X\to X, \theta \in \Theta, where X is a Polish space and \Theta is arbitrary set of indices are considerd. At x\in X, On the affine recursion on R_d+ in the critical case • Mathematics • 2021 We fix d ≥ 2 and denote S the semi-group of d× d matrices with non negative entries. We consider a sequence (An, Bn)n≥1 of i. i. d. random variables with values in S × R+ and study the asymptotic On recurrence of reflected random walk on the half-line. With an appendix on results of Martin Benda • Mathematics • 2006 Let (Y_n) be a sequence of i.i.d. real valued random variables. Reflected random walk (X_n) is defined recursively by X_0=x \ge 0, X_{n+1} = |X_n - Y_{n+1}|. In this note, we study recurrence ## References SHOWING 1-10 OF 48 REFERENCES On invariant measures of stochastic recursions in a critical case We consider an autoregressive model on \mathbb{R} defined by the recurrence equation X_n=A_nX_{n-1}+B_n, where \{(B_n,A_n)\} are i.i.d. random variables valued in \mathbb{R}\times\mathbb{R}^+ The random difference equation X\sb n=A\sb nX\sb {n-1}+B\sb n in the critical case • Mathematics • 1997 Let (B n , A n ) n≥1 be a sequence of i.i.d. random variables with values in R d x R * + . The Markov chain on R d which satisfies the random equation X n = A n X n - 1 + B n is studied when E(log A Fixed Points of the Smoothing Transform: the Boundary Case • Mathematics • 2005 Let A=(A_1,A_2,A_3,\ldots) be a random sequence of non-negative numbers that are ultimately zero with E[\sum A_i]=1 and E \left[\sum A_{i} \log A_i \right] \leq 0. The uniqueness of 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} $$where a_n\in GL(d,\mathbb{R}),b_n\in \mathbb{R}^d. Under natural Tail-homogeneity of stationary measures for some multidimensional stochastic recursions • Mathematics • 2008 We consider a stochastic recursion Xn+1 = Mn+1Xn + Qn+1, ($${n\in \mathbb {N}}$$), where (Qn, Mn) are i.i.d. random variables such that Qn are translations, Mn are similarities of the Euclidean space Fixed points of the smoothing transformation • Mathematics • 1983 SummaryLet W1,..., WN be N nonnegative random variables and let$$\mathfrak{M}$$be the class of all probability measures on [0, ∞). Define a transformation T on$$\mathfrak{M} by letting Tμ be
ASYMPTOTIC BEHAVIOR OF THE INVARIANT MEASURE FOR A DIFFUSION RELATED TO AN NA GROUP
• Mathematics
• 2005
On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup μt generated by a second order subelliptic left-invariant
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