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Given a multivariate normal distribution with an autoregressive covariance structure, for instance N(0,sigmaG(rho)), where sigma is the variance and G is a block diagonal matrix parametrized by rho (i.e. serial correlation parameter). The diagonal components of G say G_{i} are the autoregressive covariance matrices.

I am considering a transformation of this distribution to symmetric mixtures of two multivariate normal distributions having the same covariance structure, for instance

0.5*N(-0.2,sigma_{1}G(rho_{1})) + 0.5*N(0.2,sigma_{2}G(rho_{2}))

Suppose sigma=0.3 and rho=0.5. My question is how can we find the numerical values of the variance and the serial correlation parameters in the mixture distribution.

I greatly appreciate your help.

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