  We find the asymptotic distribution of the multi-dimensional multi-scale and
kernel estimators for high-frequency financial data with microstructure.
Sampling times are allowed to be asynchronous and endogenous. In the process,
we show that the classes of multi-scale and kernel estimators for smoothing
noise perturbation are asymptotically equivalent in the sense of having the
same asymptotic distribution for corresponding kernel and weight functions. The
theory leads to multi-dimensional stable central limit theorems and feasible
versions. Hence they allow to draw statistical inference for a broad class of
multivariate models which paves the way to tests and confidence intervals in
risk measurement for arbitrary portfolios composed of high-frequently observed
assets. As an application, we enhance the approach to construct a test for
investigating hypotheses that correlated assets are independent conditional on
a common factor.
