I'm trying to implement the Hayashi - Yoshida estimator for correlation (T. Hayashi, N. Yoshida: On covariance estimation of non-synchronously observed diffusion processes, 2005) and there's something I'm missing with respect to realized correlation and realized variance. Assuming that $Y = \log P$, and that I have access to $N$ discrete (asynchronous) observations for two assets on a time grid $[0,t]$, the HY estimator is as follows:
$$RV^{(i)}_{[0,t]}=\sum_{\tau \in[0,t]}\left(Y^{(i)}_{\tau}-Y^{(i)}_{\tau-1}\right)^2$$
$$RC^{(1,2)}_{[0,t]}=\frac{\sum_{\tau_1 \in[0,t]}\sum_{\tau_2 \in[0,t]}\left(Y^{(1)}_{\tau_1}-Y^{(1)}_{\tau_2-1}\right)\left(Y^{(2)}_{\tau_2}-Y^{(2)}_{\tau_2-1}\right)\mathbb{I}_{[\tau_1-1, \tau_1]\cap[\tau_2-1, \tau_2]}}{\sqrt{RV^{(1)}_{[0,t]}\cdot RV^{(2)}_{[0,t]}}}$$
where $RV$ estimates the realized variance and $RC$ the realized correlation over the period. Now, both these estimators should be unbiased but I was wondering, how do they relate exactly to the variance and correlation (Pearson)? I found empirically that $\sqrt{RV/n}\approx \sigma$ (sample std), which I guess makes sense (I'm not completely sure why), but I can't seem to find any similar scaling relationship between $RC$ and $\rho$.
The data is tick-by-tick over one day (not sure if it's relevant).
