[Affine stochastic recursions and stable laws]
We consider a multivariate affine stochastic recursion and the corresponding Birkhoff sum along a trajectory. Under a condition on the law of coefficients which is generic, we show that the above sum, suitably normalized, converges in distribution to a stable law, depending essentially on the multiplicative part of the relation. The proof is based on the spectral properties of the associated Markov operator, and on the homogeneity at infinity of the stationary measure.
Nous considérons une relation de récurrence affine multivariée à coefficients aléatoires et la somme de Birkhoff correspondante le long dʼune trajectoire. Sous une condition générique pour la loi des coefficients, nous montrons que cette somme, convenablement normalisée, converge en distribution vers une loi stable, qui dépend essentiellement de la partie multiplicative de la relation. La preuve est basée sur les propriétés spectrales de lʼopérateur de Markov associé et lʼhomogénéité à lʼinfini de la mesure stationnaire.
Accepted:
Published online:
Zhiqiang Gao 1; Yves Guivarcʼh 2; Émile Le Page 3
@article{CRMATH_2013__351_1-2_69_0, author = {Zhiqiang Gao and Yves Guivarc'h and \'Emile Le Page}, title = {Relations de r\'ecurrence \`a coefficients al\'eatoires et lois stables}, journal = {Comptes Rendus. Math\'ematique}, pages = {69--72}, publisher = {Elsevier}, volume = {351}, number = {1-2}, year = {2013}, doi = {10.1016/j.crma.2013.01.002}, language = {fr}, }
TY - JOUR AU - Zhiqiang Gao AU - Yves Guivarcʼh AU - Émile Le Page TI - Relations de récurrence à coefficients aléatoires et lois stables JO - Comptes Rendus. Mathématique PY - 2013 SP - 69 EP - 72 VL - 351 IS - 1-2 PB - Elsevier DO - 10.1016/j.crma.2013.01.002 LA - fr ID - CRMATH_2013__351_1-2_69_0 ER -
Zhiqiang Gao; Yves Guivarcʼh; Émile Le Page. Relations de récurrence à coefficients aléatoires et lois stables. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 69-72. doi : 10.1016/j.crma.2013.01.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.01.002/
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☆ The project is partially supported by National Nature Science Foundation of China (Grant Nos. 11101039, 11271045) and the Research Fund for the Doctoral Program of Higher Education (Grant No. 20100003110004).
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