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.

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