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Math Help - need help

  1. #1
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    amman/ jordan
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    need help

    hi all i need the proof of the following question but i dont know from where i get it please help me
    1. if xn converge to x uniformly then xn converge to x in r-th mean
    2. if xn converge to x almost sure then xn converge to x in probability
    3.if xn converge to x every where then xn converge to x almost sur
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  2. #2
    Member Focus's Avatar
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    Quote Originally Posted by starmh View Post
    hi all i need the proof of the following question but i dont know from where i get it please help me
    1. if xn converge to x uniformly then xn converge to x in r-th mean
    2. if xn converge to x almost sure then xn converge to x in probability
    3.if xn converge to x every where then xn converge to x almost sur
    1. I believe that reverse Fatou's lemma should do the trick.

    2. Consider the event  \{ |X_n-X|>\epsilon \mbox{  i.o.}\} which is a superset of \{|X_n-X|>\epsilon\}. Show that this set is disjoint from the set \{ \omega: \lim X_n(\omega)=X(\omega)\}.

    3. What is this set equal to \{\omega: \lim X_n(\omega) \neq X(\omega)\} ? When you figure it out, the measure of it will become obvious.
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  3. #3
    MHF Contributor matheagle's Avatar
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    Most books cover the various modes of convergence...
    Probability theory: independence ... - Google Books
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