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Math Help - tight measure

  1. #1
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    tight measure

    Hello people, I'm needing some help here.

    Is it correct to say that a family of probability measure with finite p-moment is tight?

    I saw this result only for gaussians (for p = 2, Billingsley, Probability and Measure, 3ed, page 337).

    For example, let Xn be a r.v with finite p-moment. So, by Chebychev inequality,

    Pn({x : |x| > c) \leq E|Xn|^p / c^p < K/c^p.

    Therefore, taking sup_n in both sides, for a large c,

    Pn({x : |x| > c) < \epsilon

    and so the family {Pn} is tight. Is it correct?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by gustavodecastro View Post
    Hello people, I'm needing some help here.

    Is it correct to say that a family of probability measure with finite p-moment is tight?
    Let me rephrase it: "Any family of probability measures with bounded p-moment is tight. "
    And your proof is correct. This is true for any p\geq 1.
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  3. #3
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    Thank you, Laurent!
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