Let $X_{ij}$ be independent (but not identically distributed) real-valued random variables for $1\leq i\leq j\leq n.$ Let $X$ be the symmetric matrix whose entries are given by $X_{ij}.$ Let \begin{eqnarray} A & = \big|\big|X\big|\big|~, \\ B & = \mathbb{E}\big|\big|X\big|\big|~, \\ C & = \big|\big|\mathbb{E}X\big|\big|~. \end{eqnarray}

Concentration of measure typically quantifies the deviation $|A-B|.$ I am interested in general results that quantify the deviation $|A-C|.$ Please make any assumptions you need. I am trying to understand the main techniques that help in this problem.

EDIT: Specifically, suppose that $X_{ij}$ are non-negative with means $0 < \mathbb{E}X_{ij}<1.$ What kinds of bounds on the tail distribution of these random variables can help prove a bound on the deviation $|A-C|.$ For instance, a bound on the variance of each $X_{ij}$ or a sub-Gaussian bound on the tail of $X_{ij}.$