Probability Theory and Stochastic Processes with Applications

This book provides an introduction to probability theory and discrete and continuous Stochastic Processes and its applications. It has a unique approach that provides a broad and wide introduction into the fascinating area of probability theory.

The introduction summarizes the book in prose and some popular pop star problems like famous paradoxa, the mathematics starts in chapter 2. All measure theory and real analysis is covered in detail. The chapter ends with the central limit theorem and the law of the iterated log. This is all pretty standard and is the material which is covered usually in a serious probability theory course. Chapter 3 covers discrete stochastic processes and Martingales. Chapter 4 covers continuous stochastic processes like Brownian motion up to stochstic differential equations. Also chapters 3 and 4 are pretty standard material and cover what is usually offered in a second graduate course. Chapter 5 is special and makes the book unique.

There are various selected topics like Percolation, random matrices, estimation theory (Rao Cramer), Vlasov dynamics, multivariate distributions, Poisson processes, random maps, circular random variables (central limit theorem for those), lattice points near brownian paths, arithmetic random variables, symmetric diophantine equations, random variables with singular continuous laws.