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- Yimin Xiao
- 2006

Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X… (More)

- DAVAR KHOSHNEVISAN, YIMIN XIAO, YUQUAN ZHONG, Y. ZHONG
- 2003

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)

We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process X in R d , 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)

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)

- 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

Sufficient conditions for a real-valued Gaussian random field X = {X(t), t ∈ R N } 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)

Let X = {X(t), t ∈ R + } be an operator stable Lévy process in R d with exponent B, where B is an invertible linear operator on R d. 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.

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)