Pratt's Lemma is : &#x03BE;<!-- ξ --> , &#x03B7;<!-- η --> , &#x03B6;<!-- ζ --> an

glycleWogry

glycleWogry

Answered question

2022-06-24

Pratt's Lemma is : ξ , η , ζ and ξ n , η n , ζ n such that:
ξ n ξ , η n η , ζ n ζ , convergence in probability
and η n ξ n ζ n , E ζ n E ζ , E η n E η, and E ζ , E η , E ξ are finite, prove :
If η n 0 ζ n , then E | ξ n ξ | 0.

I know how to prove E ξ n E ξ, but E | ξ n ξ | 0 seems not easy proved from it.
My first question is how to prove E | ξ n ξ | 0.
And I don't know why should emphasize the condition " η n 0 ζ n ", so my second question is: Is there any example that E | ξ n ξ | 0 is wrong if the condition is not fullfilled.

Answer & Explanation

jmibanezla

jmibanezla

Beginner2022-06-25Added 17 answers

Is there any example that E | ξ n ξ | 0 is wrong if the condition is not fullfilled.

Yes. Take your favorite example of centered random variables ξ n such that ξ n 0 in probability but E | ξ n | 0 (e.g. ξ n = ± n with probabilities 1 / n and 0 with probability 1 1 / n) and set η n = ζ n = ξ n .
Speaking about your first question, I would argue as follows: a sequence converges in probability iff any of its subsequences contains a subsubsequence converging almost surely, therefore, we can assume wlog that all convergences in question are almost sure. Then you just apply the Fatou lemma and conclude.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?