具有一般状态空间的递归Harris过程的中心可加泛函的偏差不等式
Deviation inequalities for centered additive functionals of recurrent Harris processes having general state space
摘要：
设$X$是连续时间上的Harris常返强马尔可夫过程，其一般波兰状态空间$E,$具有不变测度$μ.$.本文利用再生方法，对具有$μ(f) =0 $（$f$必须是荷）的良函数$f$，在正常返情形下，得到了$$P_{x} (|\int_0^tf(X_s)ds|\geq t^{\frac12 + η} \ge)$$的非渐近偏差界.我们将这些界限推广到完全零经常性的情况下，在中等偏差制度。对所有带电荷的函数$f$，我们得到了一个高斯浓度界.收敛的收敛速率用过程的确定性等价表示。证明的主要成分是Nummelin分裂在连续时间，它允许引入再生时间的过程中的一个扩大的状态空间。
英文摘要：
Let $X$ be a Harris recurrent strong Markov process in continuous time with general Polish state space $E,$ having invariant measure $μ.$ In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x} (|\int_0^tf(X_s)ds|\geq t^{\frac12 + η} \ge)$$ in the positive recurrent case, for nice functions $f$ with $μ(f) =0 $ ($f$ must be a charge). We generalize these bounds to the fully null-recurrent case in the moderate deviations regime. We obtain a Gaussian contentration bound for all functions $f$ which are a charge. The rate of convergence is expressed in terms of the deterministic equivalent of the process. The main ingredient of the proof is Nummelin splitting in continuous time which allows to introduce regeneration times for the process on an enlarged state space.
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