Generalized Slutsky’s Theorem

Another easy but useful corollary of Theorem 6.10 is the following generaliza­tion of Theorem 6.3:

Theorem 6.12: (Generalized Slutsky’s theorem) Let Xn a sequence of random vectors in Kk converging in probability to a nonrandom vector c. Let Ф„ (x) be a sequence of random functions on Kk satisfying plimn^TO supx єВ | Ф„ (x) –

Ф^)| = 0, where B is a closed and bounded subset of R containing c and Ф is a continuous nonrandom function on B. Then Фп (Xn) ^ p Ф(с).

Proof: Exercise.

This theorem can be further generalized to the case in which c = X is a random vector simply by adding the condition that P [X є B] = 1, but the current result suffices for the applications of Theorem 6.12.

This theorem plays a key role in deriving the asymptotic distribution of an M-estimator together with the central limit theorem discussed in Section 6.7.

6.4.2. The Uniform Strong Law of Large Numbers and Its Applications

The results of Theorems 6.10-6.12 also hold almost surely. See Appendix 6.B for the proofs.

Theorem 6.13: Under the conditions of Theorem 6.10, supee0|(1/n) Tnj=1 g(Xj, в) – E[g(X1, в)]|^ 0 a. s.

Theorem 6.14: Under the conditions of Theorems 6.11 and 6.13, в ^ во a. s.

Theorem 6.15: Under the conditions of Theorem 6.12 and the additional con­dition that Xn ^ c a. s., Фп(Xn) ^ Ф^) a. s.

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