2026-09-19T12:27:43Z https://opus4.kobv.de/opus4-uni-passau/oai

oai:kobv.de-opus4-uni-passau:1519 2025-08-13 bibliography:false doc-type:PhDThesis open_access ddc ddc:510

Qualitative and quantitative convergence results for randomised integration methods 0000-0002-9915-168X Julian Hofstadler Dipl.-Ing. 510 In this thesis different randomised integration methods based on either, randomised Quasi-Monte Carlo, or (adaptive) Markov chain Monte Carlo methods are studied. Depending on the underlying integration problem we show qualitative and quantitative results, which ensure the asymptotic correctness of an algorithm or provide explicit error bounds. The first problem we consider is Lebesgue integration in the unit cube. We prove that a class of structured randomised integration methods is consistent w.r.t. convergence in mean and probability for any integrable function. Under slightly stronger integrability conditions we show that one also has almost sure convergence for median modified methods. We demonstrate the applicability of our theoretical results by considering randomly shifted lattice rules, randomised (t,d)-sequences, Latin hypercube samples, and randomised Frolov points. Secondly, we study integration w.r.t. probability measures which are available only via their non-normalised density. In this context we investigate Markov chain Monte Carlo methods which satisfy a spectral gap condition and functions which do not need to have a finite second moment. We prove error bounds for the absolute mean error where the rate of convergence is optimal. Illustrative scenarios where our theory is applicable are the random walk Metropolis algorithm as well as slice samplers. Finally, we study so-called adaptive increasingly rare Markov chain Monte Carlo algorithms. Based on a simultaneous Wasserstein contraction assumption we estimate the mean squared error and also prove bounds which characterise the path-wise convergence of the estimator. To demonstrate the applicability of our results we consider a number of examples, among which are doubly intractable distributions. Universität Passau Passau Innstrasse 29, 94032 Passau Daniel Rudolf Matti Vihola 2024-12-09 2024-12-17 PhDThesis Text urn:nbn:de:bvb:739-opus4-15196 application/pdf eng Creative Commons - CC BY - Namensnennung 4.0 International thesis.doctoral Universität Passau Passau Fakultät für Informatik und Mathematik 1 https://opus4.kobv.de/opus4-uni-passau/oai/container/index/docId/1519 https://opus4.kobv.de/opus4-uni-passau/frontdoor/index/index/docId/1519