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Can I say mean and variance of the circular convolution ($X \circledast Y= \sum_{j=1}^{n} X_{i-j} Y_{j} $) of two Gaussian random variables $X\sim N(a,v)$ and $Y\sim N(b,w)$, is a Gaussian random variable with the mean ($a+b$) and variance ($v+w$)?

Niki
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    What does $x_j$ mean? Is it a copy of $X$ variable? – Jakub Bartczuk Mar 27 '18 at 17:26
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    This "circular convolution" is more simply seen to be the dot product of two vectors. It will have a decidedly non-Gaussian distribution, one that (assuming the joint distribution of all the $x_i$ and $y_j$ is multivariate Gaussian) is messy to express analytically (it is closely related to non-central $\chi^2$ distributions). See https://stats.stackexchange.com/questions/262604 for an answer to a generalization of this question. – whuber Mar 27 '18 at 17:35

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