Can anybody help me, Im trying to think of an example to show that convergence in mean does not imply convergence P a.s and also to show in the other direction, i.e convergence in P a.s does not imply convergence in mean??
Convergence a.s , Convergence in probability and convergence in mean
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measure-theory
probability-theory
convergence
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2Hint: The dominated convergence theorem tells you that an example in which a.s. convergence does not imply mean convergence must be very much unbounded. For the other direction, just take nondegenerate iid random variables. – 2012-06-07