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Limit theorems for stochastic recursions with Markov dependent coefficients

100%

Buraczewski D., Letachowicz M.

Colloquium Mathematicum

2012

tom 128

nr 2

263-276

We consider the stochastic recursion $Xₙ = AₙX_{n-1} + Bₙ$ for Markov dependent coefficients (Aₙ,Bₙ) ∈ ℝ⁺ × ℝ. We prove the central limit theorem, the local limit theorem and the renewal theorem for the partial sums Sₙ = X₁+ ⋯ + Xₙ.

Regular behavior at infinity of stationary measures of stochastic recursion on NA groups

100%

Buraczewski D., Damek E.

Colloquium Mathematicum

2010

tom 118

nr 2

499-523

Let N be a simply connected nilpotent Lie group and let $S = N ⋊ (ℝ⁺)^{d}$ be a semidirect product, $(ℝ⁺)^{d}$ acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙX_{n-1} + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that $lim_{t→∞} t^{χ₀}ν{x: |x| > t} = C > 0$. In particular, this applies to classical Poisson kernels on symmetric spaces, bounded homogeneous domains in ℂⁿ or homogeneous manifolds of negative curvature.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Buraczewski D.

1 Damek E.

1 Letachowicz M.

1 2012

1 2010

Wyniki wyszukiwania

Limit theorems for stochastic recursions with Markov dependent coefficients

Regular behavior at infinity of stationary measures of stochastic recursion on NA groups