# Finding the pdf of sum of squared weighted gaussian variables

I have 3 sets of weighted gaussians that are part of 3 different Gaussian Mixture Models, phi1,phi2 and phi3.

phi1 has n1 gaussian components with weights wj, mean mu_j and variance eta_j_squared, j ranging from 1 to n1. phi2 has n2 such components and phi3 has n3 components. Assume phi1,phi2 and phi3 are unimodal.

I need to square each mixture and sum them up. I know that sum of the square of gaussians results in a chi squared distribution.

Does the sum of squared gaussian mixtures still result in a chi squared distribution?

Any help would be appreciated. Thanks