# Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(X_i,X_j)$ the correlation between $X_i$ and $X_j$, and $\hat{\rho}(X_i,X_j)$ be the estimated correlation from a sample of size $T$. For $t > 0$, $$\mathbb{P}\left(\sup_{ij} \left|\hat{\rho}(X_i,X_j) - \rho(X_i,Y_j)\right| \geq t \right) \leq ?$$

To obtain such a bound, I first derived a tail bound for $\mathbb{P}\left(\left|\hat{\rho}(X_i,X_j) - \rho(X_i,Y_j)\right| \geq t \right)$: $$\mathbb{P}\left(\left|\hat{\rho}(X_i,X_j) - \rho(X_i,Y_j)\right| \geq t \right) \leq \exp\left\{ -\frac{1}{2}T \left(\sqrt{1 + \frac{2t}{1-\rho}} + 1 + \frac{t}{1-\rho} \right) + \ln 2 \right\}$$ Derivation can be found here on stats.stackexchange.

Then, naively I obtain: $$\mathbb{P}\left(\sup_{ij} \left|\hat{\rho}(X_i,X_j) - \rho(X_i,Y_j)\right| \geq t \right) \leq N^2 \exp\left\{ -\frac{1}{2}T \left(\sqrt{1 + \frac{2t}{1-\rho}} + 1 + \frac{t}{1-\rho} \right) + \ln 2 \right\}$$

Can we do better?

My guess is that we can leverage the correlation between the (correlation) coefficients to improve this bound. If it can help, we can assume that we have a block-diagonal correlation matrix.

I am looking for an entry point in the relevant literature, or a more precise help.