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- Linyuan Li, Yimin Xiao
- 2004

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)

- Yimin Xiao
- 2006

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)

- Davar Khoshnevisan, Yimin Xiao, Yuquan Zhong
- 2002

The primary goal of this paper is to study the range of the random field X(t) = PN j=1 Xj(tj), where X1, . . . , XN are independent Lévy processes in Rd. To cite a typical result of this paper, let us suppose that Ψi denotes the Lévy exponent of Xi for each i = 1, . . . , N . Then, under certain mild conditions, we show that a necessary and sufficient… (More)

We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process X in Rd , and a new class of energy forms and their corresponding capacities. We then apply these connections to solve two long-standing problems in the folklore of the theory of Lévy processes. First, we compute… (More)

- Yimin Xiao
- 2006

Sufficient conditions for a real-valued Gaussian random field X = {X(t), t ∈ RN} with stationary increments to be strongly locally nondeterministic are proven. As applications, small ball probability estimates, Hausdorff measure of the sample paths, sharp Hölder conditions and tail probability estimates for the local times of Gaussian random fields are… (More)

- Yimin Xiao
- 1997

- Dongsheng Wu, Yimin Xiao
- 2006

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)

Let X = {X(t), t ∈ R+} be an operator stable Lévy process in R with exponent B, where B is an invertible linear operator on R. We determine the Hausdorff dimension and the packing dimension of the range X([0, 1]) in terms of the real parts of the eigenvalues of B. Running Title Dimension of Operator Stable Lévy Processes