arXiv subjects

Ou Zhao

Publications and source records attributed to Ou Zhao.

6 recordsLinked to original sources

arXiv2022

Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes

Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metric $W_1$, where a related, integrated characteristic turns out to be optimal. As an application, we establish novel weak Edgeworth expansion and CLTs in $L^p$ and $W_1$. As another application, we show that a large class of high dimensional linear statistics admit Edgeworth expansions without any smoothness constraints, that is, no non-lattice condition or related is necessary. In all results, the necessary weak-dependence assumptions are very mild. In particular, we show that many prominent dynamical systems and models from time series analysis are within our framework, giving rise to many new results in these areas.

math.PR↗

arXiv2018

Big Data Analytics, Machine Learning and Artificial Intelligence in Next-Generation Wireless Networks

The next-generation wireless networks are evolving into very complex systems because of the very diversified service requirements, heterogeneity in applications, devices, and networks. The mobile network operators (MNOs) need to make the best use of the available resources, for example, power, spectrum, as well as infrastructures. Traditional networking approaches, i.e., reactive, centrally-managed, one-size-fits-all approaches and conventional data analysis tools that have limited capability (space and time) are not competent anymore and cannot satisfy and serve that future complex networks in terms of operation and optimization in a cost-effective way. A novel paradigm of proactive, self-aware, self- adaptive and predictive networking is much needed. The MNOs have access to large amounts of data, especially from the network and the subscribers. Systematic exploitation of the big data greatly helps in making the network smart, intelligent and facilitates cost-effective operation and optimization. In view of this, we consider a data-driven next-generation wireless network model, where the MNOs employ advanced data analytics for their networks. We discuss the data sources and strong drivers for the adoption of the data analytics and the role of machine learning, artificial intelligence in making the network intelligent in terms of being self-aware, self-adaptive, proactive and prescriptive. A set of network design and optimization schemes are presented with respect to data analytics. The paper is concluded with a discussion of challenges and benefits of adopting big data analytics and artificial intelligence in the next-generation communication system.

cs.IT↗

arXiv2010

A central limit theorem for reversible processes with non-linear growth of variance

Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a reversible Markov chain the condition $E(S_n^{2})/n \to κ\in (0,\infty)$ implies the convergence of the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, $E(S_n^{2}) = n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly non-standard) normal distribution are developed.

math.PR↗

arXiv2008

Estimating a monotone trend

Motivated by global warming issues, we consider a time se- ries that consists of a nondecreasing trend observed with station- ary fluctuations, nonparametric estimation of the trend under monotonicity assumption is considered. The rescaled isotonic es- timators at an interior point are shown to converge to Chernoff's distribution under minimal conditions on the stationary errors. Since the isotonic estimators suffer from the spiking problem at the end point, two modifications are proposed. The estima- tion errors for both estimators of the boundary point are shown to have interesting limiting distributions. Approximation accu- racies are assessed through simulations. One highlight of our treatment is the proof of the weak convergence results which involve several recent techniques developed in the study of con- ditional central limit questions. These weak convergences can be shown to hold conditionally given the starting values.

math.ST↗

arXiv2008

On martingale approximations

Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $π$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $π$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n|W_1)^2]=o(n)$ and $\lim_{n\to \infty}E(S_n^2)/n<\infty$. Assuming the first of these, let $\Vert g\Vert^2_+=\limsup_{n\to \infty}E(S_n^2)/n$; then $\Vert\cdot\Vert_+$ defines a pseudo norm on the subspace of $L^2(π)$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\Vert\cdot\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(π)$ into itself, and consider co-isometries ($QQ^*=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(π)$ is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of $g$ with respect to this basis.

math.PR↗

arXiv2007

Law of the iterated logarithm for stationary processes

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $R_n$ is a remainder term for which $E(R_n^2)=o(n)$. Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting $\Vert\cdot\Vert$ denote the norm in $L^2(P)$, a sufficient condition for the partial sums of a stationary process to have the form $S_n=M_n+R_n$ is that $n^{-3/2}\Vert E(S_n|X_0,X_{-1},...)\Vert$ be summable. A sufficient condition for the LIL is only slightly stronger, requiring $n^{-3/2}\log^{3/2}(n)\Vert E(S_n|X_0,X_{-1},...)\Vert$ to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.

math.PR↗