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Problem about almost sure and L1 convergence

  1. Nov 11, 2009 #1
    Can anyone give me some advice about this problem? Thanks.

    Let [tex]\lim_{n \to \infty}X_{n}=X[/tex] a.s.

    And let [tex]Y=\sup_n|X_{n}-X|[/tex].

    • Proove that [tex]Y<\infty[/tex] a.s.
    • Let [tex]Q[/tex] a new probability measure defined as it follows:
      [tex]\displaystyle Q(A)=\frac{1}{c} \mathbb{E}\!\left[1_{A} \frac{1}{1+Y}\right][/tex], where [tex]\displaystyle c=\mathbb{E}\!\left[\frac{1}{1+Y}\right][/tex].
      Proove that [tex]X_{n} \rightarrow X[/tex] (in [tex]L_{1}(Q)[/tex]).
    Last edited: Nov 11, 2009
  2. jcsd
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