arXiv subjects

Xinghua Zheng

Publications and source records attributed to Xinghua Zheng.

At least 19 recordsLinked to original sources

arXiv2024

Sub-Gaussian High-Dimensional Covariance Matrix Estimation under Elliptical Factor Model with 2 + εth Moment

We study the estimation of high-dimensional covariance matrices under elliptical factor models with 2 + εth moment. For such heavy-tailed data, robust estimators like the Huber-type estimator in Fan, Liu and Wang (2018) can not achieve sub-Gaussian convergence rate. In this paper, we develop an idiosyncratic-projected self-normalization (IPSN) method to remove the effect of heavy-tailed scalar parameter, and propose a robust pilot estimator for the scatter matrix that achieves the sub-Gaussian rate. We further develop an estimator of the covariance matrix and show that it achieves a faster convergence rate than the generic POET estimator in Fan, Liu and Wang (2018).

math.ST↗

arXiv2024

Tests for principal eigenvalues and eigenvectors

We establish central limit theorems for principal eigenvalues and eigenvectors under a large factor model setting, and develop two-sample tests of both principal eigenvalues and principal eigenvectors. One important application is to detect structural breaks in large factor models. Compared with existing methods for detecting structural breaks, our tests provide unique insights into the source of structural breaks because they can distinguish between individual principal eigenvalues and/or eigenvectors. We demonstrate the application by comparing the principal eigenvalues and principal eigenvectors of S\&P500 Index constituents' daily returns over different years.

math.ST↗

arXiv2022

High-dimensional covariance matrices under dynamic volatility models: asymptotics and shrinkage estimation

We study the estimation of the high-dimensional covariance matrix andits eigenvalues under dynamic volatility models. Data under such modelshave nonlinear dependency both cross-sectionally and temporally. We firstinvestigate the empirical spectral distribution (ESD) of the sample covariancematrix under scalar BEKK models and establish conditions under which thelimiting spectral distribution (LSD) is either the same as or different fromthe i.i.d. case. We then propose a time-variation adjusted (TV-adj) sample co-variance matrix and prove that its LSD follows the same Marcenko-Pasturlaw as the i.i.d. case. Based on the asymptotics of the TV-adj sample co-variance matrix, we develop a consistent population spectrum estimator and an asymptotically optimal nonlinear shrinkage estimator of the unconditionalcovariance matrix

math.ST↗

arXiv2021

Supercritical Spatial SIR Epidemics: Spreading Speed and Herd Immunity

We study supercritical spatial SIR epidemics on $\mathbb{Z}^2\times \{1,2,\ldots, N\}$, where each site in $\mathbb{Z}^2$ represents a village and $N$ stands for the village size. We establish several key asymptotic results as $N\to\infty$. In particular, we derive the probability that the epidemic will last forever if the epidemic is started by one infected individual. Moreover, conditional on that the epidemic lasts forever, we show that the epidemic spreads out linearly in all directions and derive an explicit formula for the spreading speed. Furthermore, we prove that the ultimate proportion of infection converges to a number that is constant over space and find its explicit value. An important message is that if there is no vaccination, then the ultimate proportion of population who will be infected can be \emph{much higher} than the vaccination proportion that is needed in order to prevent sustained spread of the infection.

math.PR↗

arXiv2021

On the Maximal Displacement of Near-critical Branching Random Walks

We consider a branching random walk on $\mathbb{Z}$ started by $n$ particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring $1+θ/n$. For $t\geq 0$, we study $M_{nt}$, the rightmost position reached by the branching random walk up to generation $[nt]$. Under certain moment assumptions on the branching law, we prove that $M_{nt}/\sqrt{n}$ converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of $M_{nt}$. We also confirm that when $θ>0$, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky in [28]. The rightmost position over all generations, $M:=\sup_t M_{nt}$, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when $θ<0$.

math.PR↗

arXiv2019

Testing High-dimensional Covariance Matrices under the Elliptical Distribution and Beyond

We develop tests for high-dimensional covariance matrices under a generalized elliptical model. Our tests are based on a central limit theorem (CLT) for linear spectral statistics of the sample covariance matrix based on self-normalized observations. For testing sphericity, our tests neither assume specific parametric distributions nor involve the kurtosis of data. More generally, we can test against any non-negative definite matrix that can even be not invertible. As an interesting application, we illustrate in empirical studies that our tests can be used to test uncorrelatedness among idiosyncratic returns.

math.ST↗

arXiv2019

Repositioning Bikes with Carrier Vehicles and Bike Trailers in Bike Sharing Systems

Bike Sharing Systems (BSSs) have been adopted in many major cities of the world due to traffic congestion and carbon emissions. Although there have been approaches to exploiting either bike trailers via crowdsourcing or carrier vehicles to reposition bikes in the ``right'' stations in the ``right'' time, they do not jointly consider the usage of both bike trailers and carrier vehicles. In this paper, we aim to take advantage of both bike trailers and carrier vehicles to reduce the loss of demand with regard to the crowdsourcing of bike trailers and the fuel cost of carrier vehicles. In the experiment, we exhibit that our approach outperforms baselines in several datasets from bike sharing companies.

cs.AI↗

arXiv2017

On the inference about the spectral distribution of high-dimensional covariance matrix based on high-frequency noisy observations

In practice, observations are often contaminated by noise, making the resulting sample covariance matrix a signal-plus-noise sample covariance matrix. Aiming to make inferences about the spectral distribution of the population covariance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (signal) sample covariance matrices depends on that of signal-plus-noise-type sample covariance matrices. As an application, we consider inferences about the spectral distribution of integrated covolatility (ICV) matrices of high-dimensional diffusion processes based on high-frequency data with microstructure noise. The (slightly modified) pre-averaging estimator is a signal-plus-noise sample covariance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of signal sample covariance matrices and that of the population covariance matrix, enables us to propose a two-step procedure to consistently estimate the spectral distribution of ICV for a class of diffusion processes. An alternative approach is further proposed, which possesses several desirable properties: it is more robust, it eliminates the effects of microstructure noise, and the asymptotic relationship that enables consistent estimation of the spectral distribution of ICV is the standard Marcenko-Pastur equation. The performance of the two approaches is examined via simulation studies under both synchronous and asynchronous observation settings.

math.ST↗

arXiv2016

On the Maximal Displacement of Subcritical Branching Random Walks

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $n\in\mathbb{N},$ let $M_{n}$ be the rightmost position reached by the branching random walk up to generation $n$. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $ρ>1$ such that the function \[ g(c,n):=ρ^{cn} P(M_{n}\geq cn), \quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, \] satisfies the following properties: there exist $0<\underlineδ\leq \overlineδ < {\infty}$ such that if $c<\underlineδ$, then $$ 0<\liminf_{n\rightarrow\infty} g (c,n)\leq \limsup_{n\rightarrow\infty} g (c,n) {\leq 1}, $$ while if $c>\overlineδ$, then \[ \lim_{n\rightarrow\infty} g (c,n)=0. \] Moreover, if the jump distribution has a finite right range $R$, then $\overlineδ < R$. If furthermore the jump distribution is "nearly right-continuous", then there exists $κ\in (0,1]$ such that $\lim_{n\rightarrow \infty}g(c,n)=κ$ for all $c<\underlineδ$. We also show that the tail distribution of $M:=\sup_{n\geq 0}M_{n}$, namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at $\underlineδ$). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.

math.PR↗

arXiv2015

On the inference about the spectra of high-dimensional covariance matrix based on noisy observations-with applications to integrated covolatility matrix inference in the presence of microstructure noise

In practice, observations are often contaminated by noise, making the resulting sample covariance matrix to be an information-plus-noise-type covariance matrix. Aiming to make inferences about the spectra of the underlying true covariance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (true) sample covariance matrices depends on that of information-plus-noise-type sample covariance matrices. As an application, we consider the inference about the spectra of integrated covolatility (ICV) matrices of high-dimensional diffusion processes based on high-frequency data with microstructure noise. The (slightly modified) pre-averaging estimator is an information-plus-noise-type covariance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of true sample covariance matrices and that of the population covariance matrix, enables us to propose a two-step procedure to estimate the spectral distribution of ICV for a class of diffusion processes. An alternative estimator is further proposed, which possesses two desirable properties: it eliminates the impact of microstructure noise, and its limiting spectral distribution depends only on that of the ICV through the standard Marčenko-Pastur equation. Numerical studies demonstrate that our proposed methods can be used to estimate the spectra of the underlying covariance matrix based on noisy observations.

math.ST↗

arXiv2014

A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes $X=(X_t)$ that solve the following martingale problem: for a given initial measure $X_0$, and for all smooth, compactly supported test functions $φ$, \begin{eqnarray*}X_t(φ)=X_0(φ)+\frac{1}{2}\int _0^tX_s(Δφ)\,ds+θ\int_0^tX_s(φ)\,ds\\{}-\int_0^tX_s(L_sφ)\,ds+M_t(φ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(φ)$ is a martingale with quadratic variation $[M(φ)]_t=\int_0^tX_s(φ^2)\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $θ_c(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $θ>θ_c(d)$, then the solution survives forever with positive probability, but if $θ<θ_c(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $θ$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $θ$; that is, for any compact set $K$, the process $X_t(K)=0$ eventually.

math.PR↗

arXiv2013

Volatility Inference in the Presence of Both Endogenous Time and Microstructure Noise

In this article we consider the volatility inference in the presence of both market microstructure noise and endogenous time. Estimators of the integrated volatility in such a setting are proposed, and their asymptotic properties are studied. Our proposed estimator is compared with the existing popular volatility estimators via numerical studies. The results show that our estimator can have substantially better performance when time endogeneity exists.

q-fin.ST↗

arXiv2013

Statistical Properties of Microstructure Noise

We study the estimation of moments and joint moments of microstructure noise. Estimators of arbitrary order of (joint) moments are provided, for which we establish consistency as well as central limit theorems. In particular, we provide estimators of auto-covariances and auto-correlations of the noise. Simulation studies demonstrate excellent performance of our estimators even in the presence of jumps and irregular observation times. Empirical studies reveal (moderate) positive auto-correlation of the noise for the stocks tested.

stat.ME↗

arXiv2012

Subcritical branching processes in random environment without Cramer condition

A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment $n$ given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.

math.PR↗

arXiv2012

On the estimation of integrated covariance matrices of high dimensional diffusion processes

We consider the estimation of integrated covariance (ICV) matrices of high dimensional diffusion processes based on high frequency observations. We start by studying the most commonly used estimator, the realized covariance (RCV) matrix. We show that in the high dimensional case when the dimension $p$ and the observation frequency $n$ grow in the same rate, the limiting spectral distribution (LSD) of RCV depends on the covolatility process not only through the targeting ICV, but also on how the covolatility process varies in time. We establish a Marčenko--Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marčenko--Pastur type theorem for RCV for a class $\mathcal{C}$ of diffusion processes. The results explicitly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class $\mathcal {C}$, the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Marčenko--Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.

stat.ME↗

arXiv2012

Discrete Fractal Dimensions of the Ranges of Random Walks in $\Z^d$ Associate with Random Conductances

Let X= {X_t, t \ge 0} be a continuous time random walk in an environment of i.i.d. random conductances {μ_e \in [1, \infty), e \in E_d}, where E_d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice Z^d and d\ge 3. Let R = {x \in Z^d: X_t = x for some t \ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. Furthermore, given any set A \subseteq Z^d, a criterion for A to be hit by X_t for arbitrarily large t>0 is given in terms of dim_H(A). Similar results for Bouchoud's trap model in Z^d (d \ge 3) are also proven.

math.PR↗

arXiv2010

The random conductance model with Cauchy tails

We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for $p^ω_{n^2t}(0,y)$ in [Ann. Probab. (2009). To appear], Theorem 5.14, to a result which gives uniform convergence for $p^ω_{n^2t}(x,y)$ for all $x,y$ in a ball.

math.PR↗

arXiv2010

Critical Branching Random Walks with Small Drift

We study critical branching random walks (BRWs) $U^{(n)}$ on~$\mathbb{Z}_{+}$ where for each $n$, the displacement of an offspring from its parent has drift~$2β/\sqrt{n}$ towards the origin and reflection at the origin. We prove that for any~$α>1$, conditional on survival to generation~$[n^α]$, the maximal displacement is asymptotically equivalent to $(α-1)/(4β)\sqrt{n}\log n$. We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like~$yn^α$ for some $y>0$ and $α>1$, and the particles are concentrated in~$[0,O(\sqrt{n})],$ then the measure-valued processes associated with the BRWs, under suitable scaling converge to a measure-valued process, which, at any time~$t>0,$ distributes its mass over~$\mathbb{R}_+$ like an exponential distribution.

math.PR↗