Almost sure convergence of quadratic form x'Ax

Question

Let $x_n$ be an $n\times 1$ vector of random variables, and $A_n=(a_{ij,n})$ be an $n\times n$ constant matrix. Suppose that $n^{-1}x_n'x_n$ converges almost surely to some limit as $n\rightarrow \infty$. What conditions are needed for $A_n$ in order to establish almost sure converge of $n^{-1}x_n'A_nx_n$?

Answer 1

We may as well assume $A_n$ are symmetric, since $x' A x$ depends only on $A + A'$. A sufficient condition is that the maximum and minimum eigenvalues of $A_n$ both approach the same limit as $n \to \infty$. Conversely, if this is not the case we can construct (non-random) $x_n$ such that $n^{-1} x_n' x_n$ converges but $n^{-1} x_n' A_n x_n$ does not.