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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

EN

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 → ∞.

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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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Invariant measures and the compactness of the domain

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Jabłoński M., Góra P.

Annales Polonici Mathematici

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1998

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tom 69

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nr 1

13-24

EN

We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ' and with some conditions on the variation $V_{[0,x]}(1/|τ'|)$ which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous "bounded variation" existence theorems.

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On convergence of iterates of the Frobenius-Perron operator for expanding mappings

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Jabłoński M.

Annales Polonici Mathematici

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1988

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tom 48

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nr 2

127-130

Ograniczanie wyników

2 Annales Polonici Mathematici

3 Jabłoński M.

1 Góra P.

1 1998

1 1991

1 1988

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

Invariant measures and the compactness of the domain

On convergence of iterates of the Frobenius-Perron operator for expanding mappings