# Sum of correlated non-Gaussian random variables with same distribution as individual terms

Let's say there is a sum $$s$$ of $$N$$ zero-mean correlated random variables $$\{x_i\}$$: $$s = x_1 + x_2 + \ldots +x_N,$$ where the correlations $$C_{ij} = \mathbb E[ x_i x_j]$$ are known. Assume that each random variable is complex, i.e. $$x_i \in \mathbb C$$, and that the real and imaginary parts are independent ("circularly-symmetric").

In the case where $$x_i$$ are circularly-symmetric complex normals, we know:

• The amplitude $$|x_i|$$ is Rayleigh distributed, and the phase $$\arg x_i$$ is uniformly distributed on $$[0, 2\pi)$$
• We know that $$s$$ will be from the same distribution as $$x_i$$, but with a different mean and variance, i.e. $$|s|$$ will be Rayleigh and $$\arg s$$ will be uniform.

For non-Gaussian noise, the amplitude distribution $$|x_i|$$ is often assumed to follow a distribution with heavier tails than Rayleigh, such as Weibull, K, or generalized gamma. Assume that even if the amplitude distribution is non-Rayleigh, the phase will still be uniformly distributed on $$[0, 2\pi)$$.

Is there a non-Rayleigh amplitude distribution on $$|x_i|$$ such that the sum $$s$$ defined above has the same distribution type as $$x_i$$? Or, are you aware of any cases where the distribution of $$|s|$$ is a common distribution with a closed-form density function?

The unique aspect of this reference request is that $$x_i$$ are not i.i.d., because they have nonzero correlations as defined above. For instance, this reference discusses the sum of generalized Gaussians in the context of maximal ratio combining (MRC) diversity, but they are i.i.d.

## migrated from dsp.stackexchange.comNov 10 '18 at 0:43

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