PREVIOUS SECTION: Uniform Integrability

1.4  Some useful properties

Property 1    If  eq081 are UI and eq152 are such that eq153a.s., then eq152 are UI.

Indeed, eq097  

eq155

QDE

Property 2    If eq152 are such that there exists an i.i.d. sequence eq081:

    (a) eq153 a.s.,

    (b) eq156,

then the sequence eq157, eq158, is UI.

 

Indeed,  

eq159

(i) eq160  eq083

(ii) SLLN:

 eq162

 eq109 From Lemma 2, (2), eq163 are UI.

 eq109 From Property 1, eq157 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  eq153 by the weaker one eq164

(that means: eq165  eq097 ).

In particular, if eq167  eq083 ( eq033 is the same eq083 ) and if eq107, then eq152 are UI.

 

Remark 3. Consider, instead of a sequence eq080, a family of r.v.'s eq169, where eq170 is an arbitrary set. Then one can introduce the following

Definition 2

(compare with Definition 1).

 eq169 are UI, if eq171 eq172 and, moreover,  

eq173

Then

(a) The statement and the proof of Lemma 1 will not change, if we replace "eq093" by "eq174".

(b) For eq175, the statement and the proof of Lemma 2 will not change, too.

(c) Properties 1 and 3 still will be true.

 

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