A note on necessary and sufficient conditions for proving that a random symmetric matrix converges to a given limit (Q1341370)

The author proves that if \(B_ n^{k \times k}\) is a sequence of symmetric matrices that converges in probability to some nonsingular symmetric matrix \(B\) element-wise and \(B_ 0\) is a nonsingular symmetric \(k \times k\) matrix such that \(0 < \| B_ 0 \|_ \infty \leq C < \infty\), then \(B = B_ 0\) if and only if both the trace and squared Euclidean norm of \({\mathcal D}_ n {\mathcal D}_ n^ T\) converge to \(k\) where \({\mathcal D}_ n = B_ 0^{-1} B_ n\). Examples are given to illustrate this result.