Chernoff Approximations for Nonstationary Random Walk Modeling

Abstract

In this paper we discuss two aspects of kinetic approach for time series modeling in terms of dynamical system. One method is based on the interpretation of kinetic equation for empirical distribution function density as a reduced description of statistical mechanics for appropriate dynamical system. For example, if distribution function density is satisfied to Liouville equation with some velocity, then this velocity can be treated as an average velocity of particle in phase space. The second method is based on the so-called Chernoff theorem from the group theory. According to the consequence from this theorem some iteration procedure exists for construction of group or semigroup, which is equivalent in some sense to average shift generator over the trajectory of appropriate dynamical system. Connection between these two methods enables us to construct a strict approach to nonstationary time series modeling with non-parametric estimation of statistical properties of corresponding sample distribution function. Also the notion of Chernoff-equivalent semigroup can be used for the calculation optimization procedure.