Approximation of quadratic form of normal variances, not necessarily non-negative

Asked Jul 23 '15 at 18:59

Active Jul 23 '15 at 18:59

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I am interested in approximating

$$ \sum_{i=1}^{n}{w_i u_i^2} $$

where $u_i \sim N(0,1)$ i.i.d. I have seen a few papers on approximating this sum where $w_i \ge 0$, but not in the general case. Why is this the case? Are there any known results on the general case?

asked Jul 23 '15 at 18:59

R S

Because $w_iu^2_i$ has a $\Gamma(1/2, 2w_i)$ distribution, your question is related to http://stats.stackexchange.com/questions/72479 (which deals with the case of a common scale factor and differing shape parameters). The possibility of negative $w_i$ appears to make yours a harder question. – whuber Jul 23 '15 at 19:28

Approximation of quadratic form of normal variances, not necessarily non-negative

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