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Math Help - Strong law of large number

  1. #1
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    Strong law of large number

    Suppose let (X_n)_{n\ge 1} be i.i.d. with N(1,3) random variables. How can I show that:

    \displaystyle\lim_{n\to\infty}\frac{X_1+...+X_n}{X  _1^2+...+X_n^2}=\frac{1}{4}  (a.s)

    From my understanding: \displaystyle\lim_{n\to\infty}\frac{S_n}{n}=\displ  aystyle\lim_{n\to\infty}\frac{1}{n}\displaystyle\s  um_{j=1}^{n}X_j=\mu
    Then the nominator will go to 1.. How about the denominator? Why is its expected value 4?
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  2. #2
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    Hello,

    It's not exactly the numerator that will go to 1, but it's X_1+\dots+X_n/n

    You have \displaystyle \frac{X_1+\dots+X_n}{X_1^2+\dots+X_n^2}=\frac{X_1+  \dots+X_n}{n}\cdot\frac{n}{X_1^2+\dots+X_n^2}

    For the second term, we know - also from the SLLN, that it tends to \mathbb{E}[X_1^2]. But we know that 3=Var[X_1]=\mathbb{E}[X_1^2]-(\mathbb{E}[X_1])^2, so the result should be...
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