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I came up to a point where I needed to add a $\sum_{i=0}^{100}(X_i^2)$, and the $\sum_{i=0}^{100}(X_i)$ where $X_i$ are iid $N(0,1)$. Now, I know that the first term is a chi squared RV with n degrees (n being the number of Gaussian RVs), which can be further approximated to a $N(100,200)$. Similarly, the second term is also $N(0,100). Can I say something about the statistical relationship between the two? I feel that they are uncorrelated, but not independent. Is there a way to prove their uncorrelatedness,(in)dependence?

Lastly, would the sum still be a Gaussian, and can I write the expression for its variance?

Thanks.

Atul Deshpande
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1 Answers1

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Well you can simply develop the formula: $$ \textstyle \text{cov}(\sum X_i^2,\sum X_i) = \sum \mathbb E(X_i^3) = 0. $$ Also they are not independant as $$ \textstyle E(\sum X_i^2 (\sum X_i)^2) \neq \mathbb E (\sum X_i^2) \mathbb E (\sum X_i)^2.$$ The sum is asymptotically Gaussian if you renormalize it (that is if you use that $(\sum X_i^2 - n)/\sqrt n \approx \mathcal N(0,1)$. You get from Cramer's theorem $$ \textstyle \frac 1 {\sqrt n} (\sum (X_i^2 + X_i) -n ) \overset{d}{\longrightarrow} \mathcal N( 0,3). $$

roger
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