econ483lec01asymp - 1 Asymptotic Theory independent and...

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1 Asymptotic Theory Assumption. Let f X i g be a sequence of independent and identically distributed ( iid ) random variables with EX i = ° and V ar ( X i ) = ± 2 ( ° and ± 2 are unknown) for i = 1 ; :::; n . Theorem 1. Law of Large Numbers (LLN) Under the above Assumption, 1 n n P i =1 ( X i ° ° ) ! p 0 ; as n ! 1 : Theorem 2. Central Limit Theorem (CLT) Under the above Assumption, 1 p n n P i =1 ( X i ° ° ) ! d N ° 0 ; ± 2 ± ; as n ! 1 : 1.1 Law of Large Numbers (LLN) 1 n n P i =1 ( X i ° ° ) ! p 0 1. the LHS is random and the RHS is nonrandom 2. If we add ° to both sides, we have 1 n n P i =1 X i ! p ° . 3. ! p is a convergence concept that connects a random variable and a nonrandom number. 4. The LLN says that in the limit ( n ! 1 ), the sum of random variables X i , or the sum of centered random variables ( X i ° ° ) weighted by n ° 1 will be a nonrandom number. This is a powerful result because a random variable gets closer to a constant. 1.2 Central Limit Theorem (CLT) 1 p n n P i =1 ( X i ° ° ) ! d N ° 0 ; ± 2 ± 1. both the LHS and RHS are random. however, the distribution of the LHS is unknown the distribution of the RHS is Normal with mean 0 and variance ± 2 2. If we divide both sides with ± , we have 1 p n n P i =1 X i ° ° ± !
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