Can anyone give me some advice about this problem? Thanks.

Let a.s.

And let .

- Proove that a.s.
- Let a new probability measure defined as it follows:

, where .

Proove that (in ).

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- Nov 11th 2009, 09:58 AMyavannaproblem about almost sure and L1 convergence
Can anyone give me some advice about this problem? Thanks.

Let a.s.

And let .

- Proove that a.s.
- Let a new probability measure defined as it follows:

, where .

Proove that (in ).

- Nov 11th 2009, 10:28 AMLaurent
It should be pointed out that the first question is just a question about real sequences (i.e., not probability) : prove that a convergent sequence is bounded, and explain why it is sufficient to conclude.

As for the second question, be very careful! It is not , but that you probably mean; this is very different. The only advice then is to use the bounded convergence theorem; that should do the trick pretty neatly. - Nov 11th 2009, 10:49 AMyavanna
- Nov 11th 2009, 11:40 AMLaurent
Yes. The assumption means that for in an event of probability 1, the sequence of real numbers converges. However, this convergence implies that the sequence is bounded. Thus, you have the inclusion of events : . If the first one has probability 1, then the same holds for the second one.