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 } .A sp rev iou s ly , our interest typically centers around the averages of this sequence b n ( ω )= 1 n n X t =1 Z t (1) C. Defnition 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 :L e t ¯ Z n = q n
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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 :G i v e n g : R k R l ( k,l < )andany 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 0 ± n a.s. −→ 0; c) X 0 X n a.s. −→ M , is ±nite and positive de±nite. Then β n exists a.s. for all n sufficiently large, and β n a.s. −→ β 0 . d) Proof: Since X 0 X/n a.s. −→ M , it follows from Proposi- tion 2.11 that det ( X 0 X/n ) a.s. −→ det ( M ). Because M is positive de±nite by (c), det ( M ) > 0. It follows that det ( X 0 X/n ) > 0 a.s. for all n sufficiently large, so ( X 0 X ) 1 exists for all n sufficiently large. Hence ˆ β n X 0 X
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Lecture15-2010 - Limits and the Law of Large Numbers...

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