arXiv 1201.45792012-01-22math.PRmath.STstat.TH

具有L2-spectral间隙的平稳马氏过程的极限定理

Limit theorems for stationary Markov processes with L2-spectral gap

Deborah Ferre, Loïc Hervé, James Ledoux

备注：: 35 pages Accepted(6 january 2011) for publication in Annales de l'Institut Henri Poincare - Probabilites et Statistiques
发表信息：: Annales de l'IHP - Probabilités et Statistiques 48, 2 (2012) 396-423
DOI：: 10.1214/11-AIHP413

摘要：

设$(X_t, Y_t)_{t\in T}$是离散或连续时间马尔可夫过程，其状态空间为$X \times R^d$，其中$X$是任意可测集.它的转移半群被假定为关于第二个分量是可加的，即假定$(X_t, Y_t)_{t\in T}$是一个马尔可夫可加过程。特别地，这意味着第一分量$(X_t)_{t\in T}$也是马尔可夫过程。马尔可夫随机游动或马尔可夫过程的可加泛函是马尔可夫可加过程的特殊例子。本文证明了过程$(Y_t)_{t\in T}$满足下列经典的极限定理：（a）中心极限定理，（b）局部极限定理，（c）一维Berry-Esseen定理，（d）一维一阶Edgeworth展开式，只要我们有supPDMATHABMR（0，1）\cap T：E{π，0}[ |Y_t| ^α] < 1，具有相对于独立情况的预期阶（对于（c）和（d），直到一些$\varepsilon > 0$）。对于陈述（b）和（d），也假设马尔可夫非格条件在独立的情况下。所有的结果都是在假设马尔可夫过程$(X_t)_{t\in T}$具有不变概率分布$π$，是平稳的，并具有$L^2(π)$ -谱隙性质（即，$(X_t)t\in N}$在离散时间情况下是$ρ$ -混合）的条件下得出的。简要讨论了$(X_t)_{t\in T}$非平稳的情况。作为应用，我们得到了与$ρ$ -混合马尔可夫链的M-估计的Berry-Esseen界。

英文摘要：

Let $(X_t, Y_t)_{t\in T}$ be a discrete or continuous-time Markov process with state space $X \times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{t\in T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{t\in T}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_t)_{t\in T}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup{t\in(0,1]\cap T : E{π,0}[|Y_t| ^α] < 1 with the expected order with respect to the independent case (up to some $\varepsilon > 0$ for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_t)_{t\in T}$ has an invariant probability distribution $π$, is stationary and has the $L^2(π)$-spectral gap property (that is, $(X_t)t\in N}$ is $ρ$-mixing in the discrete-time case). The case where $(X_t)_{t\in T}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $ρ$-mixing Markov chains.

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