Chapter 5. Asymptotic Approximations
5.2 Consistency
(A) Preliminaries
Convergence in probability and consistency (p.466, p.301) :
P
Xn X as n > 0, lim P (|Xn X| > ) = 0
n
qn = qn (X1 , , Xn ) : consistent estimator of q()
P
qn q() as n .
Tools regardi
Byeong U. Park
1
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Weak Convergence in Rk
Let X and Xn , n 1, be random vectors taking values in Rk . These random
vectors are allowed to be dened on dierent probability spaces. Below, for
the simplicity of notation, we denote all probability measures ass
Byeong U. Park/Seoul National University
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Uniform Convergence of Empirical Processes
Most of the materials in this note are taken from the book Empirical Processes in MEstimation by van de Geer (2000). Let X1 , X2 , . . . be independent copies of a ran
Review C : Basic Limit Theorems
1. Xn X cfw_nk cfw_n, cfw_nkj cfw_nk s.t Xnkj X
P
a.s.
n
j
Proof
Fact 1 : For an , a : real numbers(vectors),
an a cfw_nk cfw_n, cfw_nkj cfw_nk s.t. ankj a
n
j
)
(
a.s.
1
(|Xn X| < k ) = 1
Fact 2 : Xn X, i.e., P
n