A Guided Tour from Measure Theory to Random Processes

Organized into 6 chapters, this book presents 100 solved exercises in probability ranging from those of standard difficulty to more advanced problems. A good background in measure theory and a basic level knowledge of probability is needed. The aim of the book is to help students make the transition between simple and advanced probabilistic frameworks. Chapter 1 contains exercises aimed to emphasize the measure theoretical background of probability (e.g. monotone class theorem, uniform integrability, Lp-convergence, measure preserving and ergodic transformations, conditioning). Chapter 2 contains exercises related to the topic of the chapter: independence and conditioning. Chapter 3 is devoted to Gaussian variables and Gaussian vectors. Chapter 4 contains various exercises showing how to establish the distribution of functions of random vectors with a focus on families of beta and gamma variables and on stable distributions. Chapter 5 deals with various types of convergence of random variables and vectors, in particular with the convergence in law, including the convergence to Brownian motion and the convergence of empirical processes. Chapter 6 deals with deeper properties of Brownian motion and general stochastic processes. This is an advanced concept, with some exercises inspired by results recently appearing in research journals. Each exercise consists of several questions, references for preliminary and further reading, hints and comments, ending with a very detailed solution. The book is extremely useful for graduate and postgraduate students and those who want to better understand advanced probability theory.

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