  Motivated by statistical analysis of latent factor models for high-frequency
financial data, we develop sharp upper bounds for the spectral norm of the
realized covariance matrix of a high-dimensional It\^o semimartingale with
possibly infinite activity jumps. For this purpose, we develop Burkholder-Gundy
type inequalities for matrix martingales with the help of the theory of
non-commutative $L^p$ spaces. The obtained bounds are applied to estimating the
number of (relevant) common factors in a continuous-time latent factor model
from high-frequency data in the presence of weak factors.
