Chernoff-type bound for finite Markov chains.(English)Zbl 0938.60027

Let $$(X_n)$$ be an irreducible Markov chain on a finite set $$G$$ with transition matrix $$P$$ and stationary distribution $$\pi$$. Then, for any function $$f$$ and for any initial distribution $$q$$ the empirical mean $$n^{-1}\sum^n_{i= 1}f(X_i)$$ converges in probability to $$\pi f= \sum_y \pi(y)f(y)$$. The paper gives exponential bounds for $$P_q\left\{ n^{-1}\sum^n_{i= 1} f(X_i)-\pi f\geq \gamma\right\}$$. These bounds involve spectral gap of $$P$$.

MSC:

 60F10 Large deviations 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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References:

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