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Lecture 15-2007 - Lecture XIV Let represent the entire...

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Lecture XIV
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Let  ϖ  represent the entire random sequence  { Z t }.  As discussed last time, our interest  typically centers around the averages of this  sequence: ( 29 = = n t t n Z n b 1 1 ω
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Definition 2.9: Let { b n ( ϖ )} be a sequence of  real-valued random variables.  We say that  b n ( ϖ ) converges  almost surely  to  b , written if and only if there exists a real number  b  such  that ( 29 b b s a n → . . ω ( 29 [ ] 1 : = b b P n ω ω
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The probability measure  P  describes the  distribution of  ϖ  and determines the joint  distribution function for the entire sequence  { Z t }. Other common terminology is that  b n ( ϖ converges to b  with probability 1  (w.p.1) or  that  b n ( ϖ is strongly  consistent for  b .
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Example 2.10: Let  where { Z t } is a sequence of independently and  identically distributed (i.i.d.) random variables  with  E ( Z t )= μ < .  Then by the Komolgorov strong law of large  numbers (Theorem 3.1). = = n t t n Z n Z 1 1 μ → . . s a n Z
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Proposition 2.11: Given  g R k R l  ( k , l < ) and  any sequence { b n } such that  where  b n  and  b  are  k  x 1 vectors, if  g  is  continuous at  b , then b b s a n → . . ( 29 ( 29 b g b g s a n → . .
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Theorem 2.12: Suppose y = X β 0 + ε ; X ε / n   a.s.  0; X X / a.s. M , finite and positive definite. Then  β n  exists  a.s.  for all  n  sufficiently large,  and  β n a.s. β 0 .
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Proof: Since  X X / n   a.s. M , it follows from  Proposition 2.11 that det( X X / n a.s. det( M ).   Because  M  is positive definite by (iii),  det( M )>0.  It follows that det( X X / n )>0 a.s. for  all  n  sufficiently large, so ( X X / n ) -1  exists a.s.  for all  n  sufficiently large.  Hence ( 29 n y X n X X n ' ' ˆ 1 - β
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In addition, It follows from Proposition 2.11 that ( 29 n X n X X n ε β β ' ' ˆ 1 0 - + = 0 1 0 . . 0 0
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