  Several novel statistical methods have been developed to estimate large
integrated volatility matrices based on high-frequency financial data. To
investigate their asymptotic behaviors, they require a sub-Gaussian or finite
high-order moment assumption for observed log-returns, which cannot account for
the heavy tail phenomenon of stock returns. Recently, a robust estimator was
developed to handle heavy-tailed distributions with some bounded fourth-moment
assumption. However, we often observe that log-returns have heavier tail
distribution than the finite fourth-moment and that the degrees of heaviness of
tails are heterogeneous over the asset and time period. In this paper, to deal
with the heterogeneous heavy-tailed distributions, we develop an adaptive
robust integrated volatility estimator that employs pre-averaging and
truncation schemes based on jump-diffusion processes. We call this an adaptive
robust pre-averaging realized volatility (ARP) estimator. We show that the ARP
estimator has a sub-Weibull tail concentration with only finite 2$\alpha$-th
moments for any $\alpha>1$. In addition, we establish matching upper and lower
bounds to show that the ARP estimation procedure is optimal. To estimate large
integrated volatility matrices using the approximate factor model, the ARP
estimator is further regularized using the principal orthogonal complement
thresholding (POET) method. The numerical study is conducted to check the
finite sample performance of the ARP estimator.
