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The primary goal of this paper is to study the range of the random field X(t) = N j =1 X j (t j), where X 1 ,. .. , X N are independent Lévy processes in R d. To cite a typical result of this paper, let us suppose that i denotes the Lévy exponent of X i for each i = 1,. .. , N. Then, under certain mild conditions, we show that a necessary and sufficient(More)
An N-parameter Brownian sheet in R d maps a non-random compact set F in R N + to the random compact set B(F) in R d. We prove two results on the image-set B(F): (1) It has positive d-dimensional Lebesgue measure if and only if F has positive d 2-dimensional capacity. This generalizes greatly the earlier works of , then with probability one, we can find a(More)
Orey and Taylor (1974) introduced sets of " fast points " where Brownian increments are exceptionally large, F(λ) := {t ∈ [0, 1] : lim sup h→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 − λ 2 a.s. We prove that for any analytic set E ⊂ [0, 1], the supremum of all λ's for which E intersects F(λ)(More)
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)
A probability measure µ on R d is called weakly unimodal if there exists a constant κ ≥ 1 such that for all r > 0, (0.1) sup a∈R d µ(B(a, r)) ≤ κµ(B(0, r)). Here, B(a, r) denotes the ∞-ball centered at a ∈ R d with radius r > 0. In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of R d. In particular, we(More)
Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d defined by X(t) = X 1 (t),. .. , X d (t) , where X 1 ,. .. , X d are independent copies of a centered Gaussian random field X 0. Under certain general conditions on X 0 , we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X −1 (F), where F(More)