{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Uniform Convergence

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

Uniform Convergence - 1 Background Notes Proof of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online