Results 1 to 4 of 4

Math Help - Lebesgue integral & Mathematical expectation

  1. #1
    Junior Member
    Joined
    Nov 2009
    Posts
    57

    Lebesgue integral & Mathematical expectation

    Let \xi_1, \xi_2, ... a sequence of random variables on a probability space (\Omega, \mathcal{F}, \mathbb{P}) such that \mathbb{E}\xi_n^2\leq c for some constant c. Assume that \xi_n \rightarrow \xi almost surely as n\rightarrow \infty. How do you prove that \mathbb{E}\xi is finite and \mathbb{E}\xi_n\rightarrow\mathbb{E}\xi?

    I obviously thought about using the Dominated Convergence Theorem. Cauchy-Schwarz inequality ensures the expectation is finite. The problem is the sequence of expectations is dominated by a constant, which is not (necessarily) integrable over \Omega, and the Theorem cannot be applied as it is. But surely there is something I missed.

    Thanks by advance for you help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    A constant is integrable... Because the measure of \Omega is 1.
    So we have \mathbb{E}(\text{constant})=\text{constant}\cdot \mathbb{P}(\Omega)=\text{constant}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by akbar View Post
    Let \xi_1, \xi_2, ... a sequence of random variables on a probability space (\Omega, \mathcal{F}, \mathbb{P}) such that \mathbb{E}\xi_n^2\leq c for some constant c. Assume that \xi_n \rightarrow \xi almost surely as n\rightarrow \infty. How do you prove that \mathbb{E}\xi is finite and \mathbb{E}\xi_n\rightarrow\mathbb{E}\xi?

    I obviously thought about using the Dominated Convergence Theorem. Cauchy-Schwarz inequality ensures the expectation is finite. The problem is the sequence of expectations is dominated by a constant, which is not (necessarily) integrable over \Omega, and the Theorem cannot be applied as it is. But surely there is something I missed.

    Thanks by advance for you help.
    Hi again,

    Your problem is a consequence of this one, which is more general. In fact, in your case, the proof I gave in the mentioned thread can me much simplified as follows.

    The key word is "Egoroff": the convergence is uniform on an arbitrary large event. If the convergence is uniform, then the expectations converge. If where there is not convergence, you can use Cauchy-Schwarz to bound the expectation by a small number.

    I let you decrypt the explicitation of the previous sentence: E[\xi_n]=E[\xi_n 1_A]+E[\xi_n 1_{A^c}], E[\xi_n 1_A]\to_n E[\xi 1_A] and |E[\xi_n 1_{A^c}]|\leq c^{1/2}\epsilon^{1/2}. "qed".
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Nov 2009
    Posts
    57
    Sorry about the silly remark on the integrability of a constant. For some obscure reason I've been wrongly thinking in terms of the Lebesgue measure. Probably because I was desperate. But I was also suggesting that the result was not necessarily dependent on the choice of measure (true apparently).

    The Egoroff theorem is indeed mentioned in the book's chapter, so the proof makes perfectly sense. It was finally more tricky than I thought.

    Nice one!
    Last edited by akbar; November 19th 2009 at 03:06 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. mathematical expectation
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: May 23rd 2010, 04:18 AM
  2. another mathematical expectation ?
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 22nd 2010, 02:23 PM
  3. Mathematical Expectation
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: February 19th 2010, 03:03 PM
  4. Mathematical Expectation
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: October 26th 2009, 08:59 PM
  5. Mathematical Expectation
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: February 1st 2009, 02:26 AM

Search Tags


/mathhelpforum @mathhelpforum