Approximating a Sequence of Observation by a Simple Process

Given a sequence (s0, s1, . . . , sN ) of observations from a finite set S, we construct a process (sn) that satisfies the following properties: (i) (sn) is a piecewise Markov chain, (ii) the conditional distribution of sn given s0, . . . , sn−1 is close to the empirical transition given by the observed sequence, for most n, (iii) under (sn), with high probability the empirical frequency of the realized sequence is close to the one given by the observed sequence. We generalize this result to the case that the conditional distribution of sn given s0, . . . , sn−1 is required to be in some polyhedron Vsn−1 . ∗Laboratoire d’Analyse Géométrie et Applications, Institut Galilée, Université Paris Nord, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France. e-mail: [email protected] †MEDS Department, Kellogg School of Management, Northwestern University, and the School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. e-mail: [email protected] ‡Département Finance et Economie, HEC, 1, rue de la Libération, 78 351 Jouy-enJosas, France. e-mail: [email protected] §We thank Ehud Lehrer, Sean Meyn and Laurent Saloff-Coste for their suggestions and references. We acknowledge the financial support of the Arc-en-Ciel/Keshet program for 2001/2002. The research of the second author was supported by the Israel Science Foundation (grant No. 03620191).