PREVIOUS SECTION: Uniform Integrability
Indeed,
QDE
Property 2 | If are such that
there exists an i.i.d. sequence :
(a) a.s., (b) , then the sequence , , is UI. |
Indeed,
(i)
(ii) SLLN:
From Property 1, are UI.
QDE
Property 3 | Since the UI property is the property of
"marginal" distributions only, one can replace the
a.s.-inequality in Property 1
by the weaker one (that means: ). In particular, if ( is the same ) and if , then are UI. |
Remark 3. Consider, instead of a sequence , a family of r.v.'s , where is an arbitrary set. Then one can introduce the following
Definition 2 (compare with Definition 1). |
are UI, if and, moreover, |
Then
(a) The statement and the proof of Lemma 1 will not change, if we replace "" by "".
(b) For , the statement and the proof of Lemma 2 will not change, too.
(c) Properties 1 and 3 still will be true.
NEXT SECTION: Coupling Inequality, Maximal Coupling, Dobrushin's Theorem
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