Consider sequences of random variables ( X n ), ( Y n ) with | X n | &lt; Y n . Suppose ( Y n ) converges in probability to Y, and E Y n → E Y. I have to show that, if ( X n ) converges to X in probability, E X n → E X. I have no idea of where to start. I have seen a question here in stackexchange that answers this for the case where ( Y n ) is constant, and would like some tips on how to do this. Can anyone help me?

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Consider sequences of random variables   (      X    n    ),  (      Y    n    ) with       |        X    n        |    &amp;lt;      Y    n  . Suppose   (      Y    n    ) converges in probability to   Y, and   E      Y    n    →  E  Y. I have to show that, if   (      X    n    ) converges to   X in probability,   E      X    n    →  E  X. I have no idea of where to start. I have seen a question here in stackexchange that answers this for the case where   (      Y    n    ) is constant, and would like some tips on how to do this. Can anyone help me?

Haggar72 · Accepted Answer

Proof sketch: The claim is a well known generalization of the dominated convergence theorem if we replace convergence in measure with a.e. convergence. Note the famous result that a sequence       y    n   in a topological space   X converges to   y if and only if every subsequence of       y    n   has a further subsequence that converges to   y. Now apply this result to show that   E  (      X    n    )  →  E  (  X  ). Use the fact that convergence in measure implies convergence of a subsequence a.e.

Consider sequences of random variables ( X n </msub> ) , ( Y

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