The Central Limit Theorem for Random Dynamical Systems

Horbacz, Katarzyna

doi:10.1007/s10955-016-1601-1

Cited by 6 publications

(3 citation statements)

References 36 publications

(38 reference statements)

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“…As proved by Komorowski and Walczuk [18], exponential ergodicity implies the CLT, but they use the stronger Wasserstein norm. Recent results [14], [16] establish the CLT and exponential ergodicity simultaneously, using methods similar to ours. It is thus reasonable to expect that a general theorem in the spirit of Komorowski and Walczuk and involving the Fortet-Mourier norm can be formulated.…”

Section: Introductionmentioning

confidence: 84%

Random iteration with place dependent probabilities

Kapica

Ślęczka

2020

PMS

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We consider Markov chains arising from random iteration of functions S θ : X → X, θ ∈ Θ, where X is a Polish space and Θ is an arbitrary set of indices. At x ∈ X, θ is sampled from a distribution ϑx on Θ, and the ϑx are different for different x. Exponential convergence to a unique invariant measure is proved. This result is applied to the case of random affine transformations on R d , giving the existence of exponentially attractive perpetuities with place dependent probabilities.

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Section: Introductionmentioning

confidence: 84%

Random iteration with place dependent probabilities

Kapica

Ślęczka

2020

PMS

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“…Recently, numerous mathematicians work on central limit theorem (CLT); see, e.g., [21,18,33]. Since moderate deviation principle (MDP) fills the gap between CLT scale and LDP scale, it has been gained much attention.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

2018

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In this paper, employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.

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“…The proof also appeals to the results of M. Maxwell and M. Woodroof [18], which make it more concise and less technical than the classical proofs, based directly on martingale methods. The proofs in [9] and [10] are carried out in the same spirit, although only for some specific cases. It is also worth mentioning here that conditions proposed in this paper (namely hypothesis formulated in Sections 2 and 3) yield the Donsker invariance principle for the CLT (cf.…”

Section: Introductionmentioning

confidence: 99%

A useful version of the central limit theorem for a general class of Markov chains

Czapla

Horbacz

Wojewódka–Ściążko

2020

Journal of Mathematical Analysis and Applications

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In the paper we propose certain conditions, relatively easy to verify, which ensure the central limit theorem for some general class of Markov chains. To justify the usefulness of our criterion, we further verify it for a particular discrete-time Markov dynamical system. From the application point of view, the examined system provides a useful tool in analysing the stochastic dynamics of gene expression in prokaryotes.Obviously, V is a Lyapunov function on X 2 , which, due to (B1), satisfies the inequality U V (x, y) ≤ aV (x, y) + 2b for any (x, y) ∈ X 2 . Consequently, referring to Lemma 1.3, we can choose λ ∈ (0, 1) and c λ > 0 so that(2.9)Define T : (X 2 ) N 0 → (X 2 ) N 0 by T ((x n , y n ) n∈N 0 ) = (x n+1 , y n+1 ) n∈N 0 , and letwhere (F k ) k∈N 0 stands for the natural filtration of (φk ) k∈N 0 . Now, put Λ := max{λ, γ}. Using the fact that ρ N K ≤ ρ N J + ρ K • T ρ N J , and, further, applying sequentially the strong Markov property, condition (B5) and inequality (2.9) with m = N , we obtain

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The Central Limit Theorem for Random Dynamical Systems

Abstract: We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. The Central Limit Theorem for random dynamical systems is established.

Cited by 6 publications

References 36 publications

Random iteration with place dependent probabilities

Random iteration with place dependent probabilities

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

A useful version of the central limit theorem for a general class of Markov chains

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