Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. (English) Zbl 1190.60019

Bercu, Bernard; De Saporta, Benoîte; Gégout-Petit, Anne

Electron. J. Probab. 14, 2492-2526 (2009).

Summary: We study the asymptotic behavior of the least squares estimators of the unknown parameters of general \(p\)th-order bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem. All our analysis relies on non-standard asymptotic results for martingales.

MSC:

60F15	Strong limit theorems
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G42	Martingales with discrete parameter
62F12	Asymptotic properties of parametric estimators

Keywords:

bifurcating autoregressive process; tree-indexed times series; martingales; least squares estimation; almost sure convergence; quadratic strong law; central limit theorem