Introduction to the Mathematical and Statistical Foundations of Econometrics
The Uniform Law of Large Numbers and Its Applications
6.4.1. The Uniform Weak Law of Large Numbers
In econometrics we often have to deal with means of random functions. A random function is a function that is a random variable for each fixed value of its argument. More precisely,
Definition 6.4: Let {^, P} be the probability space. A randomfunction f (в)
on a subset © of a Euclidean space is a mapping f (ш, в): ^ x © ^ К such that for each Borel setB in К and each в є ©, {ш є ^ : f (ш, в) є B }є Ж.
Usually random functions take the form of a function g(X, в) of a random vector X and a nonrandom vector в. For such functions we can extend the weak law of large numbers for i. i.d. random variables to a uniform weak law of large numbers (UWLLN):
Theorem 6.10: (UWLLN). Let Xj, j = 1,...,n be a random sample from a k-variate distribution, and let в є © be nonrandom vectors in a closed and bounded (hence compact4) subset © c Rm. Moreover, let g(x, в) be a Borel-measurable function on Kk x © such that for each x, g(x, в) is a continuous function on ©. Finally, assume that E^^в^|g(Xj, в)|] < то. Then Plimn^<x, suPвє©|(1/п)Еn=1 g(Xj, в) - E[g(X1, в)]| = °.
Proof: See Appendix 6.A.
