Let $T_n$ be a sequence of square random matrices with $T_n$ lower triangular with $diag(T_n)=(1,1...,1)$ and $S_n$ a sequence of deterministic symmetric psd matrices. All matrices are in $R^{d\times d}$ where $d$ is fixed. Assume $$E[\|T_nS_nT_n^\top-E[T_nS_nT_n^\top]\|_F^2]\le 1/n.$$ and further assume that, with probability one, $\|T_n\|\le 1, \|S_n\|\le 1$ in operator norm. Is it always true that there exists a deterministic $\bar T_n$ lower triangular such that $$ (T_n-\bar T_n)S_n \to^P 0 $$ where $\to^P$ indicates convergence in probability?
Partial answer:
Since all matrices are bounded, by extracting a subsequence if necessary, we may assume that $E[T_nS_nT_n^\top]$ converges to some psd matrix, say $A$. The situation is significantly easier if $A$ is positive definite. Then if $L_n L_n^\top$ is the Cholesky decompoistion of $S_n$, by continuity of the Cholesky factor and the continuous mapping theorem, we have $T_nL_n \to^P \text{Chol}(A)$. We may then set $\bar T_n = \text{Chol}(A)L_n^{-1}$ and since $\|L_n\|_{op}^2=\|S_n\|_{op}\le 1$, $$ (T_n-\bar T_n)S_n = (T_n L_n - \text{Chol}(A))L_n^\top \to^P 0. $$ The more tricky issue is when $A$ is not invertible.