Cxx is the covariance matrix of a stationary random process X[k]. Assume we create X0 by subtracting the nonzero mean mx from X, consisting of samples of X[k]. Then, the covariance matrix for X0 is different from Cxx?
What if X[k] is wide-sense-stationary or not stationary?
No. Actually the covariance matrix of $X[k]$ and $X[k]+a$ is the same for a WSS process, since $$E\Bigg\{\Big(x[k+k']-E\{x[k+k']\}\Big)\cdot \Big(x^*[k]-E^*\{x[k]\}\Big)\Bigg\}\\=E\Bigg\{\Big(x[k+k']+a-E\{x[k+k']+a\}\Big)\cdot \Big(x^*[k]+a^*-E^*\{x[k]+a\}\Big)\Bigg\}$$Also a covariance matrix cannot be defined for a non-WSS process since it will be dependent on time (or $k$ here).