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A central limit theorem for processes generated by a family of transformations

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Jabłoński M.

Instytut Matematyczny Polskiej Akademii Nauk

Let ${τ_n,n≥0}$ be a sequence of measure preserving transformations of a probability space (Ω,Σ,P) into itself and let ${f_n,n≥0}$ be a sequence of elements of $L^2(Ω,Σ,P)$ with $E{f_n}=0$. It is shown that the distribution of $(∑_{i=0}^{n}f_i∘τ_i∘...∘τ_0)(D(∑_{i=0}^nf_i∘τ_i∘...∘τ_0))^{-1}$ tends to the normal distribution N(0,1) as n → ∞.

CONTENTS 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. A central limit theorem for martingale differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3. Stationary family of processes and central limit theorems for its elements. . . . . . . . . . . . . . .16 4. Central limit theorems for processes determined by endomorphisms. . . . . . . . . . . . . . . . . . 23 5. The central limit theorems for automorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 6. Final remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61

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1 Jabłoński M.

1 1991

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A central limit theorem for processes generated by a family of transformations