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Math Help - question abt notation: d_{+} or d_{-} and the Central Limit Theorem statement

  1. #1
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    question abt notation: d_{+} or d_{-} and the Central Limit Theorem statement

    The text I am reading uses some notation that I am not sure about. The only other place I saw that was in denoting limit from the left/right. Here I am not sure whether it means the same or something else.

    quoting

    The Central Limit Theorem (Bingham and Kiesel Section 2.9) tells us that

    Pr_Q[M_T\leq-d_+]->\Phi(-d_+)=1-\Phi(d_+)
    ie
    Pr_Q[M_T\geq-d_+]->\Phi(d_+)
    and similarly
    Pr_Q[M_T\geq-d_-]->\Phi(d_-)

    end quote

    I don't have the above textbook and I've seen more than one version of Central Limit Theorem. So, my questions,

    (1) is the above simply saying that the distribution of M_T converges to N(0,1)?
    (2) is the notation d_+ etc is to denote right/left continuity?

    thanks
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  2. #2
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    Hello,
    Quote Originally Posted by Volga View Post
    The text I am reading uses some notation that I am not sure about. The only other place I saw that was in denoting limit from the left/right. Here I am not sure whether it means the same or something else.

    quoting

    The Central Limit Theorem (Bingham and Kiesel Section 2.9) tells us that

    Pr_Q[M_T\leq-d_+]->\Phi(-d_+)=1-\Phi(d_+)
    ie
    Pr_Q[M_T\geq-d_+]->\Phi(d_+)
    and similarly
    Pr_Q[M_T\geq-d_-]->\Phi(d_-)

    end quote

    I don't have the above textbook and I've seen more than one version of Central Limit Theorem. So, my questions,

    (1) is the above simply saying that the distribution of M_T converges to N(0,1)?
    (2) is the notation d_+ etc is to denote right/left continuity?

    thanks
    (2) limit to the right, not necessarily continuity. This is because cumulative density functions are not always continuous at certain points and thus the probabilities can have different values. In general it doesn't really matter, but this is quite a formal (and good) way to write it

    (1) not that 'simply', but that's a consequence, yep.
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