Asymptotic Analysis for Bifurcating Autoregressive Processes via a martingale approach

Anne Gégout-Petit (1, 2) , Benoîte de Saporta (1, 2, 3) , Bernard Bercu (1, 2)

1 CQFD - Quality control and dynamic reliability
2 IMB - Institut de Mathématiques de Bordeaux
3 GREThA - Groupe de Recherche en Economie Théorique et Appliquée

We study the least-square (LS) estimator of the unknown parameters of a bifurcating auto-regressive process (BAR). Under very weak assumptions on the noise sequence (namely conditional pair-wise independence and moments of order $4$), we derive a precise rate of convergence for the LS estimator, as well as a quadratic strong law and a central limit theorem. Our main tool is martingale theory. However, standard results do not apply directly, as the martingales involved here have a special form and an exponential growth rate.

Mots clés: Bifurcating auto-regression; Tree-indexed times series; Martingales; Least-squares estimator; Almost sure convergence; Convergence rate; Quadratic strong law; Central limit theorem

Domaines: Statistiques [math.ST]

Anne Gégout-Petit, Benoîte de Saporta, Bernard Bercu. Asymptotic Analysis for Bifurcating Autoregressive Processes via a martingale approach. Joint Meeting of the Statistical Society of Canada and the Société Française de Statistique, May 2008, Ottawa, Canada. ⟨hal-00325866⟩