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Bounds for sums of random variables over a Markov chain

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  • Giesbrecht, N.

Abstract

In the paper bounds are introduced for operators appearing when summing up random variables over a non-homogeneous Markov chain. These bounds lead to exponential bounds for characteristic functions. They enable us to extend Esseen's upper bounds for spherical concentration functions to the case of summands defined on a Markov chain. Furthermore, they lead to sufficient conditions for distribution non-degeneracy and provide lower bounds for variances of the considered sums. The results of the paper seem also new for Markov chains with a finite number of states and for homogeneous Markov chains. The bounds introduced in the paper still hold for triangular arrays of random variables. They make it possible to prove local limit theorems in the cases where the distributions of corresponding sums tend to some infinitely divisible law.

Suggested Citation

  • Giesbrecht, N., 1994. "Bounds for sums of random variables over a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 53(2), pages 269-283, October.
    • Handle: RePEc:eee:spapps:v:53:y:1994:i:2:p:269-283
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    Cited by:

    1. Yeor Hafouta & Yuri Kifer, 2016. "A Nonconventional Local Limit Theorem," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1524-1553, December.

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