Lecture 15-2007 - Limits and the Law of Large Numbers...

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Limits and the Law of Large Numbers Lecture XV I. Almost Sure Convergence A. White, Halbert. Asymptotic Theory for Econometricians (New York: Academic Press, 1984). Chapter II. B. Let represent the entire random sequence   t Z . As discussed last time, our interest typically centers around the averages of this sequence:   n t t n Z n b 1 1 C. Definition 2.9: Let     n b be a sequence of real-valued random variables. We say that   n b converges almost surely to b , written   b b s a n  . . if and only if there exists a real number b such that     1 : b b P n . 1. The probability measure P describes the distribution of and determines the joint distribution function for the entire sequence   t Z . 2. Other common terminology is that   n b converges to b with probability 1 (w.p.1) or that   n b is strongly consistent for b . 3. Example 2.10: Let n t t n Z n Z 1 1 where   t Z is a sequence of independently and identically distributed (i.i.d.) random variables with   t EZ   . Then . . s a n Z by the Komolgorov strong law of large numbers (Theorem 3.1). 4. Proposition 2.11: Given   :, kl g R R k l   and any sequence   n b such that b b s a n . . where n b and b are 1 k vectors, if g is continuous at b , then
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AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XV Professor Charles B. Moss Fall 2007 2     b g b g s a n  . . . 5. Theorem 2.12: Suppose a) 0 yX   ; b) .. 0 as X n  ; c) XX M n , finite and positive definite. Then n exists a.s. for all n sufficiently large, and 0 n  . d) Proof: Since X X n M , it follows from Proposition 2.11 that     det det X X n M . Because M is positive definite by (c),   det 0 M . It follows that   det 0 X X n a.s. for all n sufficiently large, so   1 X X n exists a.s. for all n sufficiently large. Hence   n y X n X X n ' ' ˆ 1 exists for all n sufficiently large. In addition,   1 0 ˆ ' ' n X nn It follows from Proposition 2.11 that 0 1 0 . . 0 0
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Lecture 15-2007 - Limits and the Law of Large Numbers...

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