A note on necessary and sufficient conditions for proving that a random symmetric matrix converges to a given limit

We demonstrate that if Bk x kn is a sequence of symmetric matrices that converges in probability to some fixed but unspecified nonsingular symmetric matrix B elementwise, then B = B0 for a specified matrix B0 if and only if both the trace and squared Euclidean norm of DnDTn converge to k, where Dn = B-10 Bn. Examples are given to demonstrate how this result may be used to construct hypothesis tests for the equality of covariance matrices and for model misspecification.