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Math Help - Comparison of stochastic processes at a random time

  1. #1
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    Comparison of stochastic processes at a random time

    Let X_t and Y_t, t \in \mathbb{N}, be two discrete-time Markov chains such that Pr(X_t\ge x) \le Pr(Y_t\ge x), for all x. Let T be a non-negative random variable in \mathbb{N} (T may depend on X and Y).

    Is it true that Pr(X_t\ge T) \le Pr(Y_t\ge T) ?

    Thanks
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  2. #2
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    Hint: Condition on the value of T
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  3. #3
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    Thanks, you're right, after conditioning we found that that is true.

    My only concern was that with martingales, for a stopping time T, it is not true that its expectation is zero at T (which is random) even though is zero for all fixed t>0... This does not apply in my case maybe. I have to study more theory.
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