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Thread: continuity of probability distribution

  1. #1
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    continuity of probability distribution

    would you help me how I can prove this sentence? please.
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    Last edited by rough19; Feb 24th 2010 at 10:59 PM.
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  2. #2
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    Hello,

    For the last inequality $\displaystyle P(\liminf_n A_n)\leq \liminf_n P(A_n)$ :

    Consider $\displaystyle B_N=\bigcap_{n=N}^\infty A_n$
    The sequence $\displaystyle (B_N)$ is increasing.
    So $\displaystyle P(\lim_N B_N)=\lim_N P(B_N)$ (basic property of a measure/probability)

    But $\displaystyle \lim_N B_N=\liminf_n A_n$

    So we have $\displaystyle P(\liminf_n A_n)=\lim_N P(B_N)$

    But $\displaystyle \forall N,B_N \subseteq A_N$. Thus $\displaystyle P(B_N)\leq P(A_n)$

    $\displaystyle \Rightarrow \lim_N P(B_N)=\liminf_N P(B_N) \leq \liminf_n P(A_n)$


    Finally, $\displaystyle P(\liminf_n A_n)\leq \liminf P(A_n)$


    (note : $\displaystyle \liminf \equiv \underline{\lim}$)



    Does this help ?


    For the first inequality, consider taking the compliment of the last inequality.
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  3. #3
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    Absolutely.

    Thank you so much.
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