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Math Help - help with convergence in probability

  1. #1
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    Exclamation help with convergence in probability

    If Xn->X (in probability) and Yn->Y(in probability) and Cn=Xn+Yn and C=X+Y show that Cn->C (in probability).

    what i got so far
    |Xn-X|<=ε/2 and |Yn-Y|<=ε/2
    |(Xn+Yn)-(X+Y)|=|Xn-X+Yn-Y|<=|Xn-X|+|Yn-Y|<=ε/2+ε/2=ε


    |(Xn+Yn)-(X+Y)|<=ε
    |Cn-C|<=ε
    so lim(n->inf) P(|Cn-C|>=ε)=0

    I want to verify if this is correct, and if not, what's the mistake, or if I am missing anything.

    Thanks
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  2. #2
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    The convergence in probability does not say |X_n-X|<\epsilon, what you have is for each epsilon>0, \mathbb{P}(|X_n-X|>\epsilon)\rightarrow 0 as n increases.

    The way you approach this problem is in terms of sets. What can you say about \{\omega:|X_n(\omega)+Y_n(\omega)-X(\omega)-Y(\omega)|>\epsilon\} in terms of \{\omega: |X_n(\omega)-X(\omega)|> \epsilon/2\}\cup\{\omega: |Y_n(\omega)-Y(\omega)|> \epsilon/2\}?
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  3. #3
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    {w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
    {w:|Cn(w)-C(w)|>=ε}
    P(|Cn-C|>=ε)->0
    so lim(n->inf) P(|Cn-C|>=ε)=0

    Is that right?
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  4. #4
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    Quote Originally Posted by Sneaky View Post
    {w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
    {w:|Cn(w)-C(w)|>=ε}
    P(|Cn-C|>=ε)->0
    so lim(n->inf) P(|Cn-C|>=ε)=0

    Is that right?
    It does converge but why? Why do you have the third line? You want to bound \mathbb{P}(|C_n-C|>\epsilon). Read what I have written before and use \mathbb{P}(A) \leq \mathbb{P}(B) whenever A \subset B.
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  5. #5
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    {w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
    {w:|Cn(w)-C(w)|>=ε}

    so

    {w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
    contains
    {w:|Xn(w)+X(w)|>=ε/2}U{w:|Yn(w)+Y(w)|>=ε/2}

    so

    P(|Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε) >= P(|Xn(w)+X(w)|>=ε/2)
    P(|Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε) >= P(|Yn(w)+Y(w)|>=ε/2)

    P(|Cn-C|>=ε) >= P(|Xn(w)+X(w)|>=ε/2)
    P(|Cn-C|>=ε) >= P(|Yn(w)+Y(w)|>=ε/2)

    Am I on the right track here?
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  6. #6
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    Quote Originally Posted by Sneaky View Post
    {w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
    contains
    {w:|Xn(w)+X(w)|>=ε/2}U{w:|Yn(w)+Y(w)|>=ε/2}
    If you have two events such that "If A, then B", that tells you A \subset B.
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