Abstract
We construct an absolutely regular stationary random sequence which is an instantaneous bounded function of an aperiodic recurrent Markov chain with a countable state space, such that the large deviation principle fails for the arithmetic means of the sequence, while the exponential convergence holds. We also show that exponential convergence holds for the arithmetic means of a vector valued strictly stationary bounded ϕ-mixing sequence.
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Bryc, W., Smolenski, W. On the convergence of averages of mixing sequences. J Theor Probab 6, 473–483 (1993). https://doi.org/10.1007/BF01066713
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DOI: https://doi.org/10.1007/BF01066713