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 probabilityand 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.

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Pratt&#039;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 probabilityand       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&#039;t know why should emphasize the condition &quot;      η    n    ≤  0  ≤      ζ    n  &quot;, so my second question is: Is there any example that   E      |        ξ    n    −  ξ      |    →  0 is wrong if the condition is not fullfilled.

jmibanezla · Accepted Answer

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.

Pratt's Lemma is : ξ , η , ζ an

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