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We have a sequence of random variables $x_1, x_2, x_3,...$ that are independent and are $N(0, 1/n)$ random variables. We want to show that $(x_1)^2 + (x_2)^2 + (x_3)^2 +...$ converges in probability to 1.

I have tried using Borel-Cantelli Lemma, but I am unsuccessful. Then, a lightbulb clicked, and I thought maybe the $x_n$ are converging to a delta function, so this may be the reason why. However, I cannot prove this rigorously either. This is mildly frustration; how does one approach these problems, and how do you show this?

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    You should clarify what exactly $n$ is in $N(0,1/n)$. It seems some answers have assumed $x_i$ is $N(0,1/i)$ and some have assumed you have $x_1, \dots x_n$ all iid $N(0,1/n)$.2012-05-17
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    I think it is now clear that the Xi are iid N(0, 1/n). What was not clear to me is whether 1/n in the notation represents the variance or the standard deviation and the difference is important. In my second answer it did appear that Xi was N(0,1/i). Consequently my answer would not apply if Var(Xi)=1/n. But the problem has an odd formulation because as you increase n you not only add a new term but you also change the distribution of each of the previous Xis by reducing their variance by a factor of n/(n+1). So we are not extending a fixed set of numbers.2012-05-18
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    Sasha's proof seems to assume on the one hand that the Xi are fixed random variables but on the other hand by assuming their variance is 1/n they are changing with increasing n. This is an inconsistency.2012-05-18

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