Let be a sequence of
real-valued r.v.'s.
Definition 1 | R.v.'s are uniformly integrable (UI), if and, moreover,
|
We can assume the upper bound to be monotone and right-continuous.
Lemma 1 | The following are equivalent: (i) are UI; (ii) a function : (a) ; ; ; (b) |
Note: is not essential!
Proof.
as
For , put
and, for , put . From get .
Note: is an interval, and if is its left boundary point, then . Therefore,
.
Remark 1 | As a corollary, one can get the following:
if , then from Lemma 1 such that . "If the first moment" " something more". |
Remark 2. In (2), the condition may be weakened in a natural way.
Problem No.2. How? |
But it cannot be eliminated.
Example.
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, but are not UI! |
Proof of Lemma 2.
First, note that both statements (1) and (2) are "marginal", i.e. only marginal distributions are involved. So, we can construct a coupling: .
Prove (1).
(a) Assume that : (this is a special case of UI).
Then and, ,
Therefore, .
Since , then
Then, and
(c) Prove: . Indeed,
(d) , choose : and .
Then,
Since and , then
and for any
QDE
Prove (2).
Use (b) from the proof of (1): for a given ,
Therefore, :
Now,
Set . Then
Therefore,
QDE
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