Minimax rates for the covariance estimation of multi-dimensional Lévy processes with high-frequency data
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This article studies nonparametric methods to estimate the co-integrated volatility for multi-dimensional Lévy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of semimartingales. Given n observations of increments over intervals of length 1/n, the rates of convergence are 1 / √(n) if r ≤ 1 and (n n)^(r-2)/2 if r>1 , which are optimal in a minimax sense. We bound the co-jump index activity from below with the harmonic mean. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.
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