On the speed of convergence of discrete Pickands constants to continuous ones

Abstract

In this manuscript, we address open questions raised by Dieker & Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants H δ α using a family of estimators ξ δ α (T ), T > 0, where α ∈ (0, 2] is the Hurst parameter, and δ ≥ 0 is the step-size of the regular discretization grid. We derive an upper bound for the discretization error H 0 α − H δ α , whose rate of convergence agrees with Conjecture 1 of Dieker & Yakir (2014) in case α ∈ (0, 1] and agrees up to logarithmic terms for α ∈ (1, 2). Moreover, we show that all moments of ξ δ α (T ) are uniformly bounded in (δ, T ) ∈ [0, 1] × [1, ∞] and that, as T becomes large, |E(ξ δ α (T )) − H δ α | decays no slower than exp{−CT α }.