I'm dealing with a problem that I never met before, in which the question statement is clean & brief, but not sure about the skills/theorem that I should rely on, may be LLN, CLT; or maybe other fields in statistics or mathematics. Here is the problem statement:
- Suppose you have a sequence of square random matrix $\{M_n\} \in \mathbb R^{n \times n}$, in which $n = 1, 2, 3, ....$ and a known vector $\alpha \in \mathbb R^n$. Please give the mild conditions on $M_n$ such that \begin{align} \frac{\alpha^T M_n \alpha}{\alpha^T E[M_n] \alpha} \overset{p}{\rightarrow} 1 \end{align}
I mean never met, is here $M_n$ refer to the sequence of matrices, instead of a real random variable, in other word, if $M_n \in \mathbb R$, then I could simply say that $M_n / E[M_n] \overset{p}{\rightarrow} 1$ is a sufficient condition, but for matrices, what are the corresponding conditions?