Inequalities for the Total Variation between the Distributions of a Sequence and Its Translate

Let $\xi=(\xi_k)_{k\in\mathbf{N}^*}$ be a stationary homogeneous Markov chain and its translate $\xi+a=(\xi_k+a_k)_{k\in\mathbf{N}^*}$ be a real sequence. We prove an inequality for the total variation between the distributions of~$\xi$ and~$\xi+a$. This result allows us to give sufficient condition...

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Bibliographic Details
Published in:Theory of probability and its applications 2000-07, Vol.44 (3), p.561-569
Main Author: Noquet, C.
Format: Article
Language:English
Subjects:
Inequality
Markov analysis
Random variables
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Summary:Let $\xi=(\xi_k)_{k\in\mathbf{N}^*}$ be a stationary homogeneous Markov chain and its translate $\xi+a=(\xi_k+a_k)_{k\in\mathbf{N}^*}$ be a real sequence. We prove an inequality for the total variation between the distributions of~$\xi$ and~$\xi+a$. This result allows us to give sufficient conditions for absolute continuity of these distributions. Next, we consider $\xi=(\xi_k)_{k\in\mathbf{N}^*}$ a sequence of independent and identically distributed random variables and another sequence of independent variables $\eta=(\eta_k)_{k\in\mathbf{N}^*}$, which is independent of~$\xi$. We estimate the total variation between the distributions of~$\xi$ and $\xi+\eta$ and apply the obtained results to the problem of absolute continuity.
ISSN:0040-585X
1095-7219
DOI:10.1137/S0040585X97977811