The asymptotic theories of semiparametric and nonparametric econometrics heavily depend on several statistical concepts. These include probability theory, convergence, and conditioning. Accordingly, the objective of this appendix is to present results and definitions related to these concepts that are useful for the asymptotic theory covered in this book. To do so it is assumed that the reader has a basic knowledge of probability and statistics.

Probability Concepts

The axiomatic approach to probability theory is based on the concept of a probability space and an associated probability triple (Ω, F, P) defined below:

1. (i) The Sample Space, Ω

The set Ω is called the sample space. This contains the set of all possible outcomes (elementary events) of a random experiment E. For example, consider the experiment of tossing a fair coin twice. Then the sample space is

where ω1 = HT etc. Note that ωi, i = 1, …, 4, are the outcomes or elementary events.

1. (ii) The Class of Subsets of the Sample Space, F

Consider a nonempty collection of subsets (events) of Ω, F, satisfying the following two properties:

1. (a) If a subset F1 ∈ F then ∈ F, where is the complement of the subset F1 with respect to Ω and ∈ represents “belong to.”

2. By definition the subset or event F1 is the collection of elementary events in an experiment. For example, let F1 denote the event of obtaining one head in the example (A.1). F1 therefore consists of elementary events ω1, ω2, ω3, i.e., F1 = {ω1, ω2, ω3}.

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