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We consider the estimation of nonparametric regression function with long memory data and investigate the asymptotic rates of convergence of estimators based on block thresholding. We show that the estimators achieve optimal minimax convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chip(More)
The class of anisotropic Gaussian random fields includes fractional Brownian sheets, certain operator-self-similar Gaussian random fields with stationary increments and the random string process. They arise naturally in many areas and can serve as more realistic models than fractional Brownian motion. We will discuss sample path properties of an (N,(More)
Let B = { B(t), t ∈ R+ } be an (N, d)-fractional Brownian sheet with Hurst index H = (H1, . . . , HN ) ∈ (0, 1) . The objective of the present paper is to characterize the anisotropic nature of B in terms of H. We prove the following results: (1) B is sectorially locally nondeterministic. (2) By introducing a notion of “dimension” for Borel measures and(More)
Orey and Taylor (1974) introduced sets of “fast points” where Brownian increments are exceptionally large, F(λ) := {t ∈ [0, 1] : lim suph→0 |X(t+ h) −X(t)|/ √ 2h| log h|>λ}. They proved that for λ ∈ (0, 1], the Hausdorff dimension of F(λ) is 1 − λ2 a.s. We prove that for any analytic set E ⊂ [0, 1], the supremum of all λ’s for which E intersects F(λ) a.s.(More)
A probability measure μ on Rd is called weakly unimodal if there exists a constant κ ≥ 1 such that for all r > 0, (0.1) sup a∈Rd μ(B(a, r)) ≤ κμ(B(0, r)). Here, B(a, r) denotes the `∞-ball centered at a ∈ Rd with radius r > 0. In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of Rd. In particular, we use(More)