Conference Agenda

The GARCH model is a commonly used statistical tool for describing conditional heteroskedastic processes. It has been extensively studied in both univariate and multivariate cases, and more recently, in function spaces as well. This paper builds upon the concept of the functional GARCH model, which has been defined exclusively within function spaces for each time point within the domain. This paper defines the GARCH model in general, separable Hilbert spaces, and considers the entire functions rather than point-wise definitions for the GARCH equations. The paper derives sufficient conditions for strictly stationary solutions, finite moments, and weak dependence, and discusses sufficient and necessary conditions for weak stationarity. In addition, it establishes consistent Yule-Walker estimates with explicit convergence rates for the finite-dimensional projections of the GARCH parameters and their entire representation. Finally, the usefulness of the proposed model is demonstrated through a simulation study and a real data example.

Sequential empirical processes (SEPs) are an important tool in nonparametric statistics where they are applied, for example, to change detection problems and goodness-of-fit testing. Due to this connection to applied statistics it is not only relevant to study the weak convergence of SEPs in different settings, but also to develop methods with which the distribution of possible weak limits can be approximated. To the best of my knowledge, such methods are currently lacking for function-indexed SEPs that are constructed from nonstationary time series. Addressing this problem, this talk presents work in progress on the weak convergence of multiplier SEPs for weakly dependent nonstationary arrays. A general result on the asymptotic equicontinuity of multiplier processes is established, starting from which multiplier SEPs are studied under dependency- and bracketing-conditions. Regarding the asymptotic equicontinuity, the only assumptions that need be imposed on the multiplier sequence are its independence of the data and the (uniform) existence of moments of any order. Possible extensions and statistical applications are discussed.

The eigenvectors of a spectral density matrix $\mathcal{F}(\theta)$ to a stationary Gaussian process $(X_t)_{t \in \mathbb{Z}}$ depend explicitly on the frequency $\theta \in [0,2\pi]$. The most commonly used estimator of the spectral density matrix $\mathcal{F}(\theta)$ is the smoothed periodogram, which takes the form $YY^T$ for random matrices $Y$ with independent columns that each have differing underlying covariance structure. When the covariance matrices of the columns are not simultaneously diagonalizable, such matrices $YY^T$ are out of reach for the current state of random matrix theory. In this paper, we derive a Marchenko-Pastur law in this non-simultaneously diagonalizable case. On the technical level, we make the following two contributions: 1) We introduce a generalization of graph-theoretical methods specific to Gaussian random matrices, which allow for the exploitation of independent columns without needing independent rows. 2) By means of the Lagrange inversion formula, we draw a direct connection between trace moment expansions and the Marchenko-Pastur equation.

The Marchenko-Pastur law emerges when the dimension $d$ of the process and the smoothing span $m$ of the smoothed periodogram grow at the same rate, which is slower than the number of observations $n$.