Bounds in total variation distance for discrete-time processes on the sequence space

• Main Authors: Flint, Ian, Privault, Nicolas, Torrisi, Giovanni Luca
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2020
• Subjects: Science::Mathematics; Total Variation Distance; Markov Chains
• Online Access: https://hdl.handle.net/10356/137236
• Institution: Nanyang Technological University

Let ℙ and ℙ~ be the laws of two discrete-time stochastic processes defined on the sequence space Sℕ, where S is a finite set of points. In this paper we derive a bound on the total variation distance d TV(ℙ, ℙ~) in terms of the cylindrical projections of ℙ and ℙ~. We apply the result to Markov chains with finite state space and random walks on ℤ with not necessarily independent increments, and we consider several examples. Our approach relies on the general framework of stochastic analysis for discrete-time obtuse random walks and the proof of our main result makes use of the predictable representation of multidimensional normal martingales. Along the way, we obtain a sufficient condition for the absolute continuity of ℙ~ with respect to ℙ which is of interest in its own right.