Markov Chains: Analytic and Monte Carlo Computations

Chapter 2Past, present, and future

2.1 Markov property and its extensions

2.1.1 Past -field, filtration, and translation operators

Let be a sequence of random variables. The -fields

contain all events that can be observed using exclusively , that is, the past up to time included of the sequence. Obviously

A family of nondecreasing -fields, such as , is called a filtration, and provides a mathematical framework for the accumulation of information obtained by the step-by-step observation of the sequence.

Product -field

Definition 1.2.1 gives an expression for the probability of any event ...