Gaussian Approximation Of Gaussian Scale Mixtures

For a given positive random variable V > 0 and a given Z similar to N(0, 1) independent of V, we compute the scalar t(0) such that the distance in the L-2(R) sense between ZV(1/2) and Z root t(0) is minimal. We also consider the same problem in several dimensions when V is a random positive definite matrix.