Proof of Useful Propositions as to Stochastic Convergence
Proof of Useful Propositions as to Stochastic Convergence
Proposition(a)LetfYYng be a sequence of n 1 random vectors withYYn
d
! Y.SupposefXXng is a sequence of
n 1 random vectors such that(XXn YYn)
p
! 0Then XXn converges in distribution toYY.
Proof
Denote the cumulative distribution function(cdf) ofXXn by FXXn and the cdf ofYby FY. And letZZn
def
= YYn XXn and
xbe a continuitiy point ofFY. Then
FXXn(x) = P(XXn < x) = P(YYn ZZn < x)
= P(YYn < x+ ZZn)
= P(YYn < x+ ZZn;ZZn < e)+ P(YYn < x+ ZZn;ZZ
Proof of Useful Propositions as to Stochastic Convergence
Proposition(a)LetfYYng be a sequence of n 1 random vectors withYYn
d
! Y.SupposefXXng is a sequence of
n 1 random vectors such that(XXn YYn)
p
! 0Then XXn converges in distribution toYY.
Proof
Denote the cumulative distribution function(cdf) ofXXn by FXXn and the cdf ofYby FY. And letZZn
def
= YYn XXn and
xbe a continuitiy point ofFY. Then
FXXn(x) = P(XXn < x) = P(YYn ZZn < x)
= P(YYn < x+ ZZn)
= P(YYn < x+ ZZn;ZZn < e)+ P(YYn < x+ ZZn;ZZ...
