Let x ~ CN (0,a) and y ~ CN(0,b), with a, b positive constants. Let z = x + y. I need to find the distribution of the vector t=[x; z]. My basic understanding is that vector t is also complex Gaussian distributed and it has mean [0;0]. I do not know how to compute the covariance matrix of t. Can anybody help me? Thank you.


1 Answer 1


Because x and y are independent random variables, Var(z) = Var(x) + Var(y) = a + b;

One way to compute the covariance Cov(x,z) = Cov(z,x) is to simply apply the definition of covariance.

Cov(x,z) = E[xz*]-E[x]E[z*], and here we have E[x]=E[z*]=0.

Cov(x,z) = E[x(x+y)* ]=E[xx*+xy*]=E[xx*]+E[xy*]. Note that E[xy*]=E[x]E[y*]=0.

Cov(x,z) = E[xx*].

Observe that Var(x)= E[xx*]-E[x]E[x*] = 1, and E[x]=E[x*]=0 in this exercise.

Finally, Cov(x,y) = a.

Having computed the variances and co variances it is then straightforward to write the correlation matrix.

  • $\begingroup$ These are not the correct definitions for the Complex Normal distribution. See en.wikipedia.org/wiki/…. $\endgroup$
    – whuber
    Jan 23, 2021 at 18:34
  • $\begingroup$ Isn't it so that I have used the definitions for complex normal variables, while links points to definitions for complex normal vectors? If I understand correctly, x, y and z are complex normal variables, and I compute variance and covariance for normal variables (not vectors). Am I wrong? Thanks $\endgroup$
    – Rocco
    Jan 23, 2021 at 18:50
  • $\begingroup$ You are wrong because you neglect to take complex conjugates. $\endgroup$
    – whuber
    Jan 23, 2021 at 18:51
  • 1
    $\begingroup$ OK, I see. I try to correct the answer then. $\endgroup$
    – Rocco
    Jan 23, 2021 at 19:08

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