Abstract
The purpose of this chapter is to provide a background on the results from probability and inference theory required for the study of several of the topics of contemporary econometrics.
An attempt will be made to give proofs for as many propositions as is consistent with the objectives of this chapter which are to provide the tools deemed necessary for the exposition of several topics in econometric theory; it is clearly not our objective to provide a substitute to a mathematical textbook of modern probability theory.
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Notes
- 1.
A function \(f: \quad \Omega _{1}\longrightarrow \Omega _{2}\) is said to be one-to-one if and only if, for any pair (a,b) ∈ Ω1, f(a) = f(b) implies a = b. A function \(f: \quad \Omega _{1}\longrightarrow \Omega _{2}\), where Ω i , i = 1, 2, are suitable spaces, is said to be onto if and only if for every ω 2 ∈ Ω 2 there exists ω 1 ∈ Ω 1 such that f(ω 1 ) = ω 2 .
- 2.
The function ψ, implicitly defined below, is said to be a composition function, of the functions ϕ and X, is denoted by ϕ ∘ X, and means that for any ω ∈ Ω, we first evaluate X(ω), which is an element of R and then evaluate ϕ[X(ω)]. Thus, ψ is also a measurable function transforming elements ω ∈ Ω to elements in R, thus justifying the notation \(\psi: \quad \Omega \longrightarrow R\).
- 3.
The notation a.c. means “almost certainly”; the notations a.s., read “almost surely”, or a.e., read “almost everywhere”, are also common in this connection; in this volume, however, when dealing with probability spaces we shall use invariably the term a.c.
- 4.
The term a.e. to be read almost everywhere, is a term in measure theory meaning in this case, for example, that the set \(B: x\notin A,\ \mbox{ and}\ f_{n}(x)\neq 0\) has measure zero, i.e. λ(B) = 0. Mutatis mutandis it has the same meaning as a.c., a concept we introduced when dealing with random variables in a probability space with probability measure P.
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Dhrymes, P.J. (2013). Mathematical Underpinnings of Probability Theory. In: Mathematics for Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8145-4_7
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