Nonparametric low-frequency Lévy copula estimation in a general framework

Let X be a d-dimensional Lévy process. Given the low-frequency observations $ X_t $ Xt, $ t=1,\ldots ,n $ t=1,…,n, the dependence structure of the jumps of X is estimated. In general, the Lévy measure ν describes the average jump behaviour in a time unit. Thus the aim is to estimate the dependence structure of ν by estimating the so-called Lévy copula $ \mathfrak {C} $ C of ν. We generalise known one-dimensional low-frequency techniques to construct a Lévy copula estimator $ \hat {\mathfrak {C}}_n $ Cˆn based on the above-mentioned n observations and prove $ \hat {\mathfrak {C}}_n \to \mathfrak {C} $ Cˆn→C, $ n\to \infty $ n→∞, uniformly on compact sets bounded away from zero with the rate of convergence $ \sqrt {\log n} $ log⁡n that is optimal in a certain sense. This convergence holds under quite general assumptions which also include Lévy triplets $ (\Sigma , \nu , \alpha ) $ (Σ,ν,α) with non-vanishing Brownian part $ \Sigma \neq 0 $ Σ≠0 and ν of arbitrary Blumenthal–Getoor index $ 0\le \beta \le 2 $ 0≤β≤2.