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Lecture15-2010 - Limits and the Law of Large Numbers...

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Limits and the Law of Large Numbers Charles B. Moss July 29, 2010 I. Almost Sure Convergence A. Let ω represent the entire random sequence { Z t } . As previously, our interest typically centers around the averages of this sequence b n ( ω ) = 1 n n t =1 Z t (1) C. Definition 2.9 Let { b n ( ω ) } be a sequence of real-valued ran- dom variables. WE say that b n ( ω ) converges almost surely to b , written b n ( ω ) a.s. −→ b (2) if and only if there exists a real number b such that P [ ω : b n ( ω ) b ] = 1 . (3) 1. The probability measure P describes the distribution of ω and determines the joint distribution function for the entire sequence { Z t } . 2. Other common terminology is that b n ( ω ) converges to b with probability 1 (w.p.1) or that b n ( ω ) is strongly consistent for b . 3. Example 2.10 : Let ¯ Z n = q n n i =1 Z t (4) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XV Fall 2010 where { Z t } is a sequence of independently and identically distributed (i.i.d.) random variables with E [ Z t ] = μ < . Then ¯ Z n a.s. −→ μ (5) by the Komolgorov strong law of large numbers (Theorem 3.1). 4. Proposition 2.11 : Given g : R k R l ( k, l < ) and any sequence { b n } such that b n a.s. −→ b (6) wher b n and b are k × 1 vectors, if g is continuous at b , then g ( b n ) a.s. −→ g ( b ) . (7) 5. Theorem 2.12 : Suppose a) y = 0 + ; b) X n a.s. −→ 0; c) X X n a.s. −→ M , is finite and positive definite. Then β n exists a.s. for all n sufficiently large, and β n a.s. −→ β 0 . d) Proof: Since X X/n a.s. −→ M , it follows from Proposi- tion 2.11 that det ( X X/n ) a.s. −→ det ( M ). Because M is positive definite by (c), det ( M ) > 0. It follows that det ( X X/n ) > 0 a.s. for all n sufficiently large, so ( X X ) 1 exists for all n sufficiently large. Hence ˆ β n X X n 1 X y n (8) exists for all n sufficiently large. In addition, ˆ β n = β 0 + X X n 1 X n (9) It follows from Proposition 2.11 that ˆ β n a.s.
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