Uniform Convergence

Uniform Convergence - 1 Background Notes: Proof of...

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1 Background Notes: Proof of Consistency Economics 245A There are several di&erent forms of uniform convergence of a sto- chastic function to a continuous nonstochastic function. Let g T ( ) de- note a nonnegative (to avoid the use of absolute value signs in the de±nitions with g T ( )) sequence of random variables that depend on (that is, a sequence of stochastic functions of ). (Almost Sure Uniform Convergence) If P [ lim T !1 sup 2 g T ( ) = 0] = 1 then g T ( ) converges to 0 almost surely uniformly in 2 ². Frequently the convergence is de±ned in terms of the sample average log-likelihood, which we refer to here as T 1 Q T ( ). If P [ lim T !1 sup 2 j T 1 Q T ( ) Q ( ) j = 0] = 1 then T 1 Q T ( ) converges to Q ( ) almost surely uniformly in 2 ². (Weak Semiuniform Convergence) If lim T !1 inf 2 P [ g T < " ] = 1 for any " > 0, then g T ( ) converges to 0 in probability semiuniformly in 2 ². For the sample average log-likelihood, if
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.

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Uniform Convergence - 1 Background Notes: Proof of...

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