I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. And let $(A_1,..., A_n)$ be an iid sample, s.t. $A_i \in S_{+}(p)$ for all $1 \leq i \leq n$. Furthermore, there exists a matrix $B \in S_{+}(p)$, s.t.
$\|B - \frac{1}{n}{\sum_{i = 1}^n}(B A_i)^{1/2} \| \leq \frac{C}{\sqrt{n}}$, where $\| \cdot \|$ is just some matrix norm, e.g. Frobenius.
The question is whether it is possible to state anything about the closeness of $B$ and $\frac{1}{n}\sum_{i}A_i$ in terms of sample-size? Something like
$\|B - \frac{1}{n}\sum_{i}F(A_i)\| \leq \frac{R}{\sqrt{n}}$ where $F(x)$ is a some function, e.g. $F(x) = \sqrt{x}$ or $= x^2$.