Suppose . Show in probability iff .

Any tips would be much appreciated.

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- Dec 12th 2010, 09:26 PMBeakyconvergence in probability iff expectation...
Suppose . Show in probability iff .

Any tips would be much appreciated. - Dec 13th 2010, 05:23 AMFocus
What have you done so far? Some basic hints, Markov's inequality will be useful and also splitting the expectation.

- Dec 13th 2010, 10:34 AMBeaky
I was trying with Markov's inequality earlier but didn't get anywhere. I think I've managed to show one inclusion now. I don't see how to split the expectation either which is maybe why I'm stuck on the other.

So I have expectation >> convergence since for small . - Dec 13th 2010, 02:43 PMFocus
You are supposed to show that which is pretty similar to what you did if you notice that .

For the converse try splitting as follows

You can bound the RHS term by epsilon (as I indicated above), the second term you can split further by using the fact that then use convergence in probability.