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Yimin Xiao

Publications and source records attributed to Yimin Xiao.

At least 55 records · Page 3Linked to original sources
arXiv2019

Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets

Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely convergent in all the Hölder spaces $C^γ([-M,M]^N)$, where $M>0$ and $γ\in (0, \min\{H_1,\ldots, H_N\})$ are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let $X=\{X(t), t \in \R^N\}$ be an $\R^d$-valued harmonizable fractional stable sheet whose components are independent copies of $Z^H$. By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image $X^{-1}(F)$ holds for all Borel sets $F \subseteq \R^d$. This is referred to as a uniform Hausdorff dimension result for the inverse images.

math.PR↗
arXiv2019

Cramér Type Moderate Deviations for Random Fields

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

math.ST↗
arXiv2018

Strong laws of large numbers for arrays of random variables and stable random fields

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi \cite{chobanyan-l-s} for a sequence of random variables indexed by a one-parameter.

math.PR↗
arXiv2018

Maximal Moments and Uniform Modulus of Continuity for Stable Random Fields

In this work, we solve an open problem mentioned in Xiao (2010) and provide sharp bounds on the rate of growth of maximal moments for stationary symmetric stable random fields using structure theorem of finitely generated abelian groups and ergodic theory of quasi-invariant group actions. We also investigate the relationship between this rate of growth and the path regularity properties of self-similar stable random fields with stationary increments, and establish uniform modulus of continuity of such fields. In the process, a new notion of weak effective dimension is introduced for stable random fields and is connected to maximal moments and path properties. Our results establish a boundary between shorter and longer memory in relation to Holder continuity of S alpha S random fields confirming a conjecture of Samorodnitsky (2004).

math.PR↗
arXiv2018

Sharp Space-Time Regularity of the Solution to Stochastic Heat Equation Driven by Fractional-Colored Noise

In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a Lévy process $X$ taking value in $\mathbb{R}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariance function $f$ which is the Fourier transform of a tempered measure $μ.$ After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution $\{u(t,x),\, t\ge 0,\, x\in\mathbb{R}^d\}$ in both time and space variables. Under mild conditions, we give the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function $(t,x)\mapsto u(t,x)$. Our results generalize and strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017).

math.PR↗
arXiv2018

Multiple points of operator semistable Lévy processes

We determine the Hausdorff dimension of $k$-multiple points for a symmetric operator semistable Lévy process $X=\{X(t), t\in\mathbb{R}_+\}$ in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient condition for the existence of $k$-multiple points. Our results extend to all $k\geq2$ the recent work [23], where the set of double points $(k = 2)$ was studied in the symmetric operator stable case.

math.PR↗
arXiv2018

On the favorite points of symmetric Lévy processes

This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of Lévy processes. By exploring Molchan's idea for deriving lower tail probabilities of Gaussian processes with stationary increments, we extend the result of Marcus (2001) on the favorite points to a larger class of symmetric Lévy processes.

math.PR↗
arXiv2018

Regularity properties of the solution to a stochastic heat equation driven by a fractional Gaussian noise on ${\mathbb{S}}^2$

We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

math.PR↗
arXiv2018

Spectral Conditions for Equivalence of Gaussian Random Fields with Stationary Increments

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.

math.PR↗
arXiv2017

Strong Local Nondeterminism of Spherical Fractional Brownian Motion

Let $B = \left\{ B\left( x\right),\, x\in \mathbb{S}^{2}\right\} $ be the fractional Brownian motion indexed by the unit sphere $\mathbb{S}^{2}$ with index $0<H\leq \frac{1}{2}$, introduced by Istas \cite{IstasECP05}. We establish optimal estimates for its angular power spectrum $\{d_\ell, \ell = 0, 1, 2, \ldots\}$, and then exploit its high-frequency behavior to establish the property of its strong local nondeterminism of $B$.

math.ST↗
arXiv2017

Uniform dimension results for a family of Markov processes

In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of Lévy processes, stable jump diffusion and non-symmetric stable-like processes are obtained.

math.PR↗
arXiv2017

Exact Moderate and Large Deviations for Linear Random Fields

By extending the methods in Peligrad et al. (2014a, b), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

math.PR↗
arXiv2017

Joint Asymptotics for Estimating the Fractal Indices of Bivariate Gaussian Processes

Multivariate (or vector-valued) processes are important for modeling multiple variables. The fractal indices of the components of the underlying multivariate process play a key role in characterizing the dependence structures and statistical properties of the multivariate process. In this paper, under the infill asymptotics framework, we establish joint asymptotic results for the increment-based estimators of bivariate fractal indices. Our main results quantitatively describe the effect of the cross-dependence structure on the performance of the estimators.

math.ST↗
arXiv2017

A macroscopic multifractal analysis of parabolic stochastic PDEs

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+uξ$, where $ξ$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989; 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.

math.PR↗
arXiv2016

Asymptotic behavior of semistable Lévy exponents and applications to fractal path properties

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on $\mathbb R^d$. These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar Lévy processes. The proofs are elementary, using only the properties of the Lévy exponent, and certain index formulae.

math.PR↗
arXiv2016

Hitting probabilities of random covering sets in tori and metric spaces

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.

math.PR↗