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Math Help - martingale

  1. #1
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    martingale

    a Martingale X=(X_n)_{n \geq 0} is bounded in L^2 if sup_n E(X_n ^2) \leq \infty.
    Let X be a martingale with X_n \in L^2 for each n. Show that X is bounded in L^2 iff \Sigma_{n \geq 0} ^{\infty} E((X_n-X_{n-1})^2) < \infty.

    here is what i want to know. If we know that X be a martingale with X_n \in L^2 for each n, it means that E(X_n ^2) < \infty for each n, doesnt it? so doesn't this already imply that X is bounded in L^2?
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    here is what i want to know. If we know that X be a martingale with X_n \in L^2 for each n, it means that E(X_n ^2) < \infty for each n, doesnt it? so doesn't this already imply that X is bounded in L^2?
    What if \mathbb{E}[X_n^2]=n<\infty? Then \sup E[X_n^2]=\infty.
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