Asymptotic analysis for bifurcating autoregressive processes via a martingale approach.

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

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Let $τ_{n}, n \geq 0$ be a sequence of measure preserving transformations of a probability space (Ω,Σ,P) into itself and let $f_{n}, n \geq 0$ be a sequence of elements of $L^{2} (Ω, Σ, P)$ with $E f_{n} = 0$ . It is shown that the distribution of $(\sum_{i = 0}^{n} f_{i} \circ τ_{i} \circ . . . \circ τ_{0}) {(D (\sum_{i = 0}^{n} f_{i} \circ τ_{i} \circ . . . \circ τ_{0}))}^{- 1}$ tends to the normal distribution N(0,1) as n → ∞. 1985 Mathematics Subject Classification: 58F11, 60F05, 28D99.