- Thesis:
- Bachelor in Mathematics 50%
- Author:
- Nils Tausch
- Title:
- Die Mellintransformation für multiplikative Faltungen von Wahrscheinlicheitsmaßen
- Supervisors:
- Sergio Brenner Miguel
- Jan JOHANNES
- Abstract:
- The aim of this bachelor thesis is to introduce the Mellin transform as a tool for multipli- cative convolutions of probability measures and to elaborate its main properties. For this purpose the characteristic function, which represents a comparable tool, is first considered and its central properties are proven. These include boundedness, symmetry, uniqueness, and the convolution theorem. Multiplicative convolutions and their calculation using Radon-Nikodym derivatives are then considered. With this theoretical foundation, the Mellin transform for probability distributions can then be introduced. Its most important properties can be shown with similar proof ideas as for the characteristic function afterwards: Uniqueness, convolution formula and the boundedness. Finally, the characteristic function and the Mellin transform are compared graphically.
References: - S. Brenner Miguel, F. Comte und J. Johannes. Spectral cut-off regularisation for density estimation under multiplicative measurement errors, Electronic Journal of Statistics, 15(1), 3551-3573, 2021.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |