Monday, 27 November 2023, 14:00–15:00 CET
On-site:
University of Vienna
Seminarraum 16
Kolingasse 14–16
1090 Vienna
Online:
Zoom-Link: https://univienna.zoom.us/j/68154688415?pwd=anBCaXlnZ0ZRZ0R2YkV0U2hMSXdIQT09
Meeting-ID: 681 5468 8415
Passcode: 517043
Weak dependence and optimal quantitative self-normalized central limit theorems
Abstract:
Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order $n^{-1/2}$ for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and $L^p$) distance. The results show that, in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate $n^{-1/2}$ cannot be reached, disproving a popular belief in the literature. This can be salvaged by simple methods: We reveal that in order to maintain the optimal speed of convergence $n^{-1/2}$, simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of Garch models of any order, functionals of dynamical systems arising from SDEs, iterated random functions and many more.
Bio:
Johannes Moritz Jirak has obtained a Diploma in Mathematics from the Technical University Graz before pursuing further studies abroad, earning a Master's in Recherche Probabilité et Statistiques from Université de Provence Aix-Marseille in France. He returned to the TU Graz for his Ph.D. in Mathematics. Following his education, Johannes Moritz Jirak worked at the Austrian Financial Market Supervisory Authority in Vienna. Continuing his academic journey, he served as a Postdoctoral Researcher at Humboldt University of Berlin and was a visiting scholar at the Stevanovich Center for Financial Mathematics in Chicago. Moritz Jirak's career progressed as he became a W3 Professor at the Technical University of Braunschweig. Despite receiving an offer from the University of Aarhus, he declined and, since April 2020, has held the position of Professor at the ISOR in Vienna. Johannes Moritz Jirak is currently Head of the Department of Statistics and Operations Research.
