Introduction

We introduce in this chapter those statistical and probability techniques that underlie what is presented later. Few proofs will be given because a complete treatment of even a small part of what is dealt with here would require a book in itself. We do not intend to bother the reader with too formal a presentation. We shall be concerned with a sample space, Ω, which can be thought of as the set of all conceivable realisations of the random processes with which we are concerned. If A is a subset of Ω, then P(A) is the probability that the realisation is in A. Because we deal with discrete time series almost exclusively, questions of ‘measurability’, i.e. to which sets A can P(·) be applied, do not arise and will never be mentioned. We say this once and for all so that the text will not be filled with requirements that this or that set be measurable or that this or that function be a measurable function. Of course we shall see only (part of) one realisation, {x(t), t = 0, ±1, ±2,…} and are calling into being in our mind's eye, so to say, a whole family of such realisations. Thus we might write ω (t; ω) where ω ∈ Ω is the point corresponding to a particular realisation and, as ω varies for given t, we get a random variable, i.e. function defined on the sample space Ω.