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Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

100%

Peigné M., Woess W.

Colloquium Mathematicum

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2011

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tom 125

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nr 1

55-81

EN

In this continuation of the preceding paper (Part I), we consider a sequence $(Fₙ)_{n≥0}$ of i.i.d. random Lipschitz mappings 𝖷 → 𝖷, where 𝖷 is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) $Xₙ^{x} = Fₙ ∘ ... ∘ F₁(x)$ starting at x ∈ 𝖷. The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I, the Chacon-Ornstein theorem and a hyperbolic extension of the space 𝖷 as well as the process $(Xₙ^{x})$. The results are applied to a class of examples, namely, the reflected affine stochastic recursion given by $X₀^{x} = x ≥ 0$ and $Xₙ^{x} = |AₙX_{n-1}^{x} - Bₙ|$, where (Aₙ,Bₙ) is a sequence of two-dimensional i.i.d. random variables with values in ℝ⁺⁎ × ℝ⁺⁎.

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Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

100%

Peigné M., Woess W.

Colloquium Mathematicum

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2011

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tom 125

|

nr 1

31-54

EN

Consider a proper metric space 𝖷 and a sequence $(Fₙ)_{n≥0}$ of i.i.d. random continuous mappings 𝖷 → 𝖷. It induces the stochastic dynamical system (SDS) $Xₙ^{x} = Fₙ ∘ ... ∘ F₁(x)$ starting at x ∈ 𝖷. In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic iterations. We consider the case when the Fₙ are contractions and, in particular, discuss recurrence criteria and their sharpness for the reflected random walk.

3

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

81%

Grama I., Le Page É., Peigné M.

Colloquium Mathematicum

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2014

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tom 134

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nr 1

1-55

EN

We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite-dimensional increments of the process. The distinctive feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited to processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains, our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.

Ograniczanie wyników

3 Colloquium Mathematicum

3 Peigné M.

2 Woess W.

1 Grama I.

1 Le Page É.

1 2014

2 2011

Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains