Background: Probability

Abstract

We review the fundamental tools used to establish the inferential basis for our models. Results are stated as theorems, lemmas and corollaries. Most of the key proofs are provided in Chapter 16 although, sometimes, when useful to the general development, proofs are given within the text itself. The main ideas of stochastic processes, in particular Brownian motion and functions of Brownian motion, are explained in non-measure-theoretic terms. The background to this, i.e., distribution theory and large sample results, is recalled. Rank invariance is an important concept, i.e., the ability to transform some variable, usually time, via monotonic increasing transformations without having an impact on inference. These ideas hinge on the theory of order statistics and the basic notions of this theory are recalled. An outline of the theory of counting processes and martingales is presented without leaning upon measure-theoretic constructions. The important concepts of explained variation and explained randomness are outlined in elementary terms, i.e., only with reference to random variables and, at least initially, making no explicit appeal to any particular model. This is important since the concepts are hardly any less fundamental than a concept such as variance itself. They ought therefore stand alone, and not require derivation as a particular feature of some model. In practice, of course, we may need estimate conditional distributions and making an appeal to a model at this point is quite natural.