Limit theorems for stochastically perturbed dynamical systems

Łoskot, Krzysztof; Rudnicki, Ryszard

doi:10.1017/s0021900200102906

Cited by 11 publications

(10 citation statements)

References 7 publications

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“…Ĺoskot and Rudnicki [10] consider stochastic models that satisfy the following contraction condition. (1) and ψ is the distribution of the shock , is called an average contraction on (X, ρ) if there exists a Borel function λ : X → R such that E(λ) = λ(z)ψ(dz) < 1 and…”

Section: Resultsmentioning

confidence: 99%

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Stochastic growth: asymptotic distributions

Stachurski¹

2003

Economic Theory

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This note studies conditions under which sequences of state variables generated by discrete-time stochastic optimal accumulation models have law of large numbers and central limit properties. Productivity shocks with unbounded support are considered. Instead of restrictions on the support of the shock, an “average contraction” property is required on technology. Copyright Springer-Verlag Berlin Heidelberg 2003Keywords and Phrases: Stochastic growth, Law of large numbers, Central limit theorem., JEL Classification Numbers: C51, C62, O41.,

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Section: Resultsmentioning

confidence: 99%

“…The techniques are based on results recently obtained by Ĺoskot and Rudnicki [10], who study the LLN and CLT properties of perturbed dynamical systems on Polish space. Section 2 formulates the problem and gives the major definitions.…”

Section: Introductionmentioning

confidence: 99%

Stochastic growth: asymptotic distributions

Stachurski¹

2003

Economic Theory

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“…Since EL(η 1 ) < 1 and E (x 0 , S(x 0 , η 1 )) < ∞, we infer from Theorem 1 and Remark 1 of [8] that there exists a measure µ * ∈ M 1 such that…”

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confidence: 85%

“…However, this assumption leads to time-homogeneous Markov processes. For an account of this subject we refer the reader to [8]. What we will need from this theory is a result from [8] (Theorem 1) which states, roughly speaking, that if such a system contracts on average then it has a stationary measure with finite first moment.…”

Section: S(t X) − S(t Y) ≤ Lementioning

confidence: 99%

Stochastic differential equation driven by a pure-birth process

Tyran‐Kamińska

2002

Ann. Polon. Math.

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“…It is of great interest to give sufficient conditions for the existence of an invariant probability measure for the Markov chain {Z n }, i.e. a probability measure µ * satisfyingfor all Borel sets A (see [4,9,[13][14][15][16]). …”

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confidence: 99%

Invariant measures for Markov operators with application to function systems

Szarek

2003

Studia Math.

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Abstract.A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems.0. Introduction. The theory of Markov processes is a fast developing topic which has been extensively studied during the last few years. The reason for this study was the progress in the theory of fractals. Markov processes can be considered from two points of view. They can be investigated by purely probabilistic and purely analytic methods. In our paper we use the second approach. More precisely, let {Z n } be a homogeneous Markov chain taking values in a metric space (X, ) and let π be its transition kernel, i.e. Prob(Zfor all n ∈ N and Borel sets A. It is of great interest to give sufficient conditions for the existence of an invariant probability measure for the Markov chain {Z n }, i.e. a probability measure µ * satisfyingfor all Borel sets A (see [4,9,[13][14][15][16]).In our analytic approach we consider the Markov operator P defined on the space of all Borel measures given byConsequently, searching for invariant measures of Markov processes can be replaced by searching for fixed points of P . Using this method does not involve the probabilistic background of the transition kernels π.2000 Mathematics Subject Classification: Primary 47A35; Secondary 28D15.

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Limit theorems for stochastically perturbed dynamical systems

Abstract: We consider a discrete-time stochastically perturbed dynamical system on the Polish space given by the recurrence formulaXn=S(Xn–1,Yn),whereYnare i.i.d. random elements. We prove the existence of unique stationary measure and versions of classical limit theorems for the process (Xn).

Cited by 11 publications

References 7 publications

Stochastic growth: asymptotic distributions

Stochastic growth: asymptotic distributions

Stochastic differential equation driven by a pure-birth process

Invariant measures for Markov operators with application to function systems

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