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We consider a class of stationary processes exhibiting both long-range dependence and heavy tails. Separate limit theorems for sums and for extremes have been established recently in literature with novel objects appearing in the limits. In this article, we establish the joint sum-max limit theorems for this class of processes. In the finite-variance case, the limit consists of two independent components: a fractional Brownian motion arising from the sum, and a long-range dependent random sup measure arising from the maximum. In the infinite-variance case, we obtain in the limit two dependent components: a stable process and a random sup measure whose dependence structure is described through the local time and range of a stable subordinator. For establishing the limit theorem in the latter case, we also develop a joint convergence result for the local time and range of subordinators, which may be of independent interest., Comment: 24 pages

The Internet of Things (IoT) system generates massive high-speed temporally correlated streaming data and is often connected with online inference tasks under computational or energy constraints. Online analysis of these streaming time series data often faces a trade-off between statistical efficiency and computational cost. One important approach to balance this trade-off is sampling, where only a small portion of the sample is selected for the model fitting and update. Motivated by the demands of dynamic relationship analysis of IoT system, we study the data-dependent sample selection and online inference problem for a multi-dimensional streaming time series, aiming to provide low-cost real-time analysis of high-speed power grid electricity consumption data. Inspired by D-optimality criterion in design of experiments, we propose a class of online data reduction methods that achieve an optimal sampling criterion and improve the computational efficiency of the online analysis. We show that the optimal solution amounts to a strategy that is a mixture of Bernoulli sampling and leverage score sampling. The leverage score sampling involves auxiliary estimations that have a computational advantage over recursive least squares updates. Theoretical properties of the auxiliary estimations involved are also discussed. When applied to European power grid consumption data, the proposed leverage score based sampling methods outperform the benchmark sampling method in online estimation and prediction. The general applicability of the sampling-assisted online estimation method is assessed via simulation studies., Comment: Accepted by The Annals of Applied Statistics

The recently introduced notion of tail adversarial stability has been proven useful in studying tail dependent time series and obtaining their limit theorems. Its implication and advantage over the classical strong mixing framework has been examined for max-linear processes, but not yet studied for additive linear processes that have also been commonly used in modeling extremal clusters and tail dependence in time series. In this article, we fill this gap by verifying the tail adversarial stability condition for regularly varying additive linear processes. A comparison with the classical strong mixing condition is also given in the context of tail autocorrelation estimation. We in addition consider extensions of the result on an additive linear process to its stochastic volatility generalization and to its max-linear counterpart. Some implications for limit theorems in statistical context are also discussed., Comment: 33 pages

We investigate a family of multiple-stable processes that may exhibit either long-range or short-range dependence, depending on the parameters. There are two parameters for the processes, the memory parameter $\beta\in(0,1)$ and the multiplicity parameter $p\in\mathbb N$. We investigate the macroscopic limit of extremes of the process, in terms of convergence of random sup-measures, for the full range of parameters. Our results show that (i) the extremes of the process exhibit long-range dependence when $\beta_p := p\beta-p+1\in(0,1)$, with a new family of random sup-measures arising in the limit, (ii) the extremes are of short-range dependence when $\beta_p, Comment: Major revision. 36 pages

Using the theory of generalized Weierstrass transform, we show that the Hermite rank is identical to the power rank in the Gaussian case, and that an Hermite rank higher than one is unstable with respect to a level shift., Comment: Proofs greatly simplified

We study the limit law of a vector made up of normalized sums of functions of long-range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multivariate Gaussian process involving dependent Brownian motion marginals, or (b) a multivariate process involving dependent Hermite processes as marginals, or (c) a combination. We treat cases (a), (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary.