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Math Help - Convergence of a nonnegative submartingale

  1. #1
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    Convergence of a nonnegative submartingale

    Hi, how to I show that a non-negative submartingale need not converge almost surely?
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  2. #2
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    Re: Convergence of a nonnegative submartingale

    Hey alphabeta89.

    Do sub-martingale always increase (or at minimum, remain unchanged)?
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  3. #3
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    Re: Convergence of a nonnegative submartingale

    Ya, Can I use X_n = n?
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  4. #4
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    Re: Convergence of a nonnegative submartingale

    That seems reasonable, but can you please remind what almost surely refers to?
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  5. #5
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    Re: Convergence of a nonnegative submartingale

    If X_n converge almost surely to X, then

    P(\lim_{n\to\infty}X_n = X) = 1.

    Right?
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  6. #6
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    Re: Convergence of a nonnegative submartingale

    Well then it looks OK.
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