On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case

Sara Brofferio; Dariusz Buraczewski; Ewa Damek

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 377-395
  • ISSN: 0246-0203

Abstract

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We consider the autoregressive model on ℝd defined by the stochastic recursion Xn = AnXn−1 + Bn, where {(Bn, An)} are i.i.d. random variables valued in ℝd× ℝ+. The critical case, when 𝔼 [ log A 1 ] = 0 , was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measureν for the Markov chain {Xn}. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on ℝd ∖ {0}.

How to cite

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Brofferio, Sara, Buraczewski, Dariusz, and Damek, Ewa. "On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 377-395. <http://eudml.org/doc/271983>.

@article{Brofferio2012,
abstract = {We consider the autoregressive model on ℝd defined by the stochastic recursion Xn = AnXn−1 + Bn, where \{(Bn, An)\} are i.i.d. random variables valued in ℝd× ℝ+. The critical case, when $\mathbb \{E\}[\log A_\{1\}]=0$ , was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measureν for the Markov chain \{Xn\}. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on ℝd ∖ \{0\}.},
author = {Brofferio, Sara, Buraczewski, Dariusz, Damek, Ewa},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk; random coefficients autoregressive model; affine group; random equations; contractive system; regular variation; invariant measure; homogeneous measure},
language = {eng},
number = {2},
pages = {377-395},
publisher = {Gauthier-Villars},
title = {On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case},
url = {http://eudml.org/doc/271983},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Brofferio, Sara
AU - Buraczewski, Dariusz
AU - Damek, Ewa
TI - On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 377
EP - 395
AB - We consider the autoregressive model on ℝd defined by the stochastic recursion Xn = AnXn−1 + Bn, where {(Bn, An)} are i.i.d. random variables valued in ℝd× ℝ+. The critical case, when $\mathbb {E}[\log A_{1}]=0$ , was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measureν for the Markov chain {Xn}. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on ℝd ∖ {0}.
LA - eng
KW - random walk; random coefficients autoregressive model; affine group; random equations; contractive system; regular variation; invariant measure; homogeneous measure
UR - http://eudml.org/doc/271983
ER -

References

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