Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted