Advances in modern technologies have led to an abundance of high-dimensional time series data in many fields, including finance, economics, health, engineering, and meteorology, among others. This causes the “curse of dimensionality” problem in both univariate and multivariate time series data. The main objective of time series analysis is to make inferences about the conditional distributions. There are some methods in the literature to estimate the conditional mean and conditional variance functions in time series. However, most of those are inefficient, computationally intensive, or suffer from the overparameterization. We propose some dimension reduction techniques to address the curse of dimensionality in high-dimensional time series dataFor high-dimensional matrix-valued time series data, there are a limited number of methods in the literature that can preserve the matrix structure and reduce the number of parameters significantly (Samadi, 2014, Chen et al., 2021). However, those models cannot distinguish between relevant and irrelevant information and yet suffer from the overparameterization. We propose a novel dimension reduction technique for matrix-variate time series data called the "envelope matrix autoregressive model" (EMAR), which offers substantial dimension reduction and links the mean function and the covariance matrix of the model by using the minimal reducing subspace of the covariance matrix. The proposed model can identify and remove irrelevant information and can achieve substantial efficiency gains by significantly reducing the total number of parameters. We derive the asymptotic properties of the proposed maximum likelihood estimators of the EMAR model. Extensive simulation studies and a real data analysis are conducted to corroborate our theoretical results and to illustrate the finite sample performance of the proposed EMAR model.For univariate time series, we propose sufficient dimension reduction (SDR) methods based on some integral transformation approaches that can preserve sufficient information about the response. In particular, we use the Fourier and Convolution transformation methods (FM and CM) to perform sufficient dimension reduction in univariate time series and estimate the time series central subspace (TS-CS), the time series mean subspace (TS-CMS), and the time series variance subspace (TS-CVS). Using FM and CM procedures and with some distributional assumptions, we derive candidate matrices that can fully recover the TS-CS, TS-CMS, and TS-CVS, and propose an explicit estimate of the candidate matrices. The asymptotic properties of the proposed estimators are established under both normality and non-normality assumptions. Moreover, we develop some data-drive methods to estimate the dimension of the time series central subspaces as well as the lag order. Our simulation results and real data analyses reveal that the proposed methods are not only significantly more efficient and accurate but also offer substantial computational efficiency compared to the existing methods in the literature. Moreover, we develop an R package entitled “sdrt” to easily perform our program code in FM and CM procedures to estimate suffices dimension reduction subspaces in univariate time series.

We consider the distribution of the zero and typical cells of a (homogeneous) Poisson hyperplane tessellation. We give a direct proof adapted to our setting of the well known Complementary Theorem. We provide sharp bounds for the tail distribution of the number of facets. We also improve existing bounds for the tail distribution of size measurements of the cells, such as the volume or the mean width. We improve known results about the generalised D.G. Kendall's problem, which asks about the shape of large cells. We also show that cells with many facets cannot be close to a lower dimensional convex body. We tacle the much less study problem of the number of facets and the shape of small cells. In order to obtain the results above we also develop some purely geometric tools, in particular we give new results concerning the polytopal approximation of an elongated convex body.