What is the variance of the product of a complex Gaussian matrix and vector?

[nikozm]

Hi,

Assuming that A is a n x m random matrix and each of its entries are complex Gaussian with zero mean and unit-variance. Also, assume that b is a n x1 random vector and its entries are complex Gaussian with zero mean and variance=s. Then, what would be the variance of their product Ab?

Any help would be useful.

Thanks

 

[coquelicot]

I think that you have first to know the variance of a product of two gaussians. More precisely, if X and Y are two independent random variables with distribution N(0,s) and N(0,s'), then what is the distribution of XY ? There are formulae in the litterature (Google it). Once you have obtained the distribution Z of XY, you have to know the distribution of the sum of several variable Z' of the same type (theoretically, this is the convolution product of the variables). I am almost certain that Z, and the sum of the variables Z', will have zero mean. The variance should be given in the litterature, if the sum of the Z' is a known distribution. If your matrix is large, then you can use the central limit theorem to approximate the sum of the Z'.