arXiv subjects

Yimin Xiao

Publications and source records attributed to Yimin Xiao.

At least 91 records · Page 5Linked to original sources

arXiv2011

Occupation Time Fluctuations of Weakly Degenerate Branching Systems

We establish limit theorems for re-scaled occupation time fluctuations of a sequence of branching particle systems in $\R^d$ with anisotropic space motion and weakly degenerate splitting ability. In the case of large dimensions, our limit processes lead to a new class of operator-scaling Gaussian random fields with non-stationary increments. In the intermediate and critical dimensions, the limit processes have spatial structures analogous to (but more complicated than) those arising from the critical branching particle system without degeneration considered by Bojdecki et al.{\it Stochastic Process. Appl. 2006}. Due to the weakly degenerate branching ability, temporal structures of the limit processes in all three cases are different from those obtained by Bojdecki et al. .{\it Stochastic Process. Appl. 2006}.

math.PR↗

arXiv2011

Multivariate Operator-Self-Similar Random Fields

Multivariate random fields whose distributions are invariant under operator-scalings in both time-domain and state space are studied. Such random fields are called operator-self-similar random fields and their scaling operators are characterized. Two classes of operator-self-similar stable random fields $X=\{X(t), t \in \R^d\}$ with values in $\R^m$ are constructed by utilizing homogeneous functions and stochastic integral representations.

math.PR↗

arXiv2011

Packing Dimension Profiles and Levy Processes

We extend the concept of packing dimension profiles, due to Falconer and Howroyd (1997) and Howroyd (2001), and use our extension in order to determine the packing dimension of an arbitrary image of a general Levy process.

math.PR↗

arXiv2011

$α$-Time Fractional Brownian Motion: PDE Connections and Local Times

For $0<α\leq 2$ and $0<H<1$, an $α$-time fractional Brownian motion is an iterated process $Z = \{Z(t)=W(Y(t)), t \ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $α$-stable Lévy process $\{Y(t), t\geq 0 \}$ in $\RR{R}$ independent of $\{W(t), t \in \R\}$. It is shown that such processes have natural connections to partial differential equations and, when $Y$ is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a $d$-dimensional $α$-time fractional Brownian motion $X = \{X(t), t \in \R_+$\} defined by $ X(t)=\big(X_{1}(t),..., X_{d}(t) \big), $ where $t\geq 0$ and $X_{1},..., X_{d}$ are independent copies of $Z$, are investigated. Our methods rely on the strong local ' nondeterminism of fractional Brownian motion.

math.PR↗

arXiv2011

Fractal Dimensions for Continuous Time Random Walk Limits

In a continuous time random walk (CTRW), each random jump follows a random waiting time. CTRW scaling limits are time-changed processes that model anomalous diffusion. The outer process describes particle jumps, and the non-Markovian inner process (or time change) accounts for waiting times between jumps. This paper studies fractal properties of the sample functions of a time-changed process, and establishes some general results on the Hausdorff and packing dimensions of its range and graph. Then those results are applied to CTRW scaling limits.

math.PR↗

arXiv2010

Hausdorff and packing dimensions of the images of random fields

Let $X=\{X(t),t\in\mathbb{R}^N\}$ be a random field with values in $\mathbb{R}^d$. For any finite Borel measure $μ$ and analytic set $E\subset\mathbb{R}^N$, the Hausdorff and packing dimensions of the image measure $μ_X$ and image set $X(E)$ are determined under certain mild conditions. These results are applicable to Gaussian random fields, self-similar stable random fields with stationary increments, real harmonizable fractional Lévy fields and the Rosenblatt process.

math.ST↗

arXiv2009

Fractal and Smoothness Properties of Space-Time Gaussian Models

Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts interests are often in the fractal nature of the sample surfaces and in the rate of change of the spatial surface at a given location in a given direction. In this paper we apply the theory of Yaglom (1957) to construct a large class of space-time Gaussian models with stationary increments, establish bounds on the prediction errors and determine the smoothness properties and fractal properties of this class of Gaussian models. Our results can be applied directly to analyze the stationary space-time models introduced by Cressie and Huang (1999), Gneiting (2002) and Stein (2005), respectively.

math.ST↗

arXiv2009

Uniform Modulus of Continuity of Random Fields

A sufficient condition for the uniform modulus of continuity of a random field $X = \{X(t), t \in \R^N\}$ is provided. The result is applicable to random fields with heavy-tailed distribution such as stable random fields.

math.PR↗

arXiv2009

Regularity of Intersection Local Times of Fractional Brownian Motions

Let $B^{α_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${α_i}$ ($i=1,2$), and let $B^{α_1}$ and $B^{α_2}$ be independent. We prove that, if $\frac{N_1}{α_1}+\frac{N_2}{α_2}>d$, then the intersection local times of $B^{α_1}$ and $B^{α_2}$ exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points. One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre ({\it J. Theor. Probab.} {\bf 20} (2007)), where the existence of the intersection local times of two independent $(1,d)$-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.

math.PR↗

arXiv2008

Local times of multifractional Brownian sheets

Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\in{\mathbb{R}}_+^N$ with values in $(0,1)^N$. Let $\{B^{H(t)}(t)\}=\{B^{H(t)}(t),t\in{\mathbb{R}}^N_+\}$ be an $(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$. Under some regularity conditions on the function $H(t)$, we prove the existence, joint continuity and the Hölder regularity of the local times of $\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the level sets of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.

math.PR↗

arXiv2008

A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes

Let $p \in (0, \infty)$ be a constant and let $\{ξ_n\} \subset L^p(Ω, {\mathcal F}, ¶)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} ξ_k$. It is proved that, if there exist a nondecreasing function $φ: \R_+\to \R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge 2$ such that $$ \sum_{n=0}^\infty \sup_{k \ge 0} \E\bigg|\frac{S_{k, a^n}} {φ(a^n)} \bigg|^p < \infty,$$ Then $$ \lim_{n \to \infty} \frac{S_{0, n}} {φ(n)} = 0\qquad \hbox{a.s.} $$ This extends Theorem 1 in Levental, Chobanyan and Salehi \cite{chobanyan-l-s} and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.

math.PR↗

arXiv2008

Correlated continuous time random walks

Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy tailed jumps, and the time-fractional version codes heavy tailed waiting times. This paper develops scaling limits and governing equations in the case of correlated jumps. For long-range dependent jumps, this leads to fractional Brownian motion or linear fractional stable motion, with the time parameter replaced by an inverse stable subordinator in the case of heavy tailed waiting times. These scaling limits provide an interesting class of non-Markovian, non-Gaussian self-similar processes.

math.PR↗

arXiv2008

Joint continuity of the local times of fractional Brownian sheets

Let $B^H=\{B^H(t),t\in{\mathbb{R}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

math.PR↗

arXiv2008

Linear Fractional Stable Sheets: wavelet expansion and sample path properties

In this paper we give a detailed description of the random wavelet series representation of real-valued linear fractional stable sheet introduced in Ayache, Roueff and Xiao (2007). By using this representation, in the case where the sample paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound for their behavior at infinity and around the coordinate axes. The Hausdorff dimensions of the range and graph of these stable random fields are then derived.

math.ST↗

arXiv2007

Sample Path Properties of Bifractional Brownian Motion

Let $B^{H, K}= \big\{B^{H, K}(t), t \in \R_+ \big\}$ be a bifractional Brownian motion in $\R^d$. We prove that $B^{H, K}$ is strongly locally nondeterministic. Applying this property and a stochastic integral representation of $B^{H, K}$, we establish Chung's law of the iterated logarithm for $B^{H, K}$, as well as sharp Hölder conditions and tail probability estimates for the local times of $B^{H, K}$. We also consider the existence and the regularity of the local times of multiparameter bifractional Brownian motion $B^{\bar{H}, \bar{K}}= \big\{B^{\bar{H}, \bar{K}}(t), t \in \R^N_+ \big\}$ in $\R^d$ using Wiener-Itô chaos expansion.

math.PR↗

arXiv2007

Large deviations for local time fractional Brownian motion and applications

Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0\}$ be the local time process at zero of a strictly stable Lévy process $X=\{X_t, t \ge 0\}$ of index $1<α\leq 2$ independent of $W^H$. The $\a$-stable local time fractional Brownian motion $Z^H=\{Z^H(t), t \ge 0\}$ is defined by $Z^H(t) = W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1 - \frac 1 α)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coupleCTRW,limitCTRW}). However, $Z^H$ does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process $Z^H$. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for $Z^H$.

math.PR↗

arXiv2007

Harmonic analysis of additive Levy processes

Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional Lévy processes, and consider the $N$-parameter random field \[\X(\bm{t}):= X_1(t_1)+...+X_N(t_N).\] First we demonstrate that for all nonrandom Borel sets $F\subseteq\R^d$, the Minkowski sum $\X(\R^N_+)\oplus F$, of the range $\X(\R^N_+)$ of $\X$ with $F$, can have positive $d$-dimensional Lebesgue measure if and only if a certain capacity of $F$ is positive. This improves our earlier joint effort with Yuquan Zhong \ycite{KXZ:03} by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in \fullocite{KXZ:03}, the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. We present a few new consequences here.

math.PR↗

arXiv2007

Packing-Dimension Profiles and Fractional Brownian Motion

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles ${\rm Dim}_s$ that are parametrized by real numbers $s>0$. Subsequently, Howroyd (2001) introduced alternate $s$-dimensional packing dimension profiles $\hbox{${\rm P}$-$\dim$}_s$ and proved, among many other things, that $\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all integers $s>0$ and all analytic sets $E\subseteq\R^N$. The goal of this article is to prove that $\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all real numbers $s>0$ and analytic sets $E\subseteq\R^N$. This answers a question of Howroyd (2001, p. 159). Our proof hinges on a new property of fractional Brownian motion.

math.PR↗