Billingsley P. Probability and Measure (Wiley, 1986)(ISBN 0471804789)(K)(600dpi)(T)(635s)_MV
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What is new, then, is the alternation of probability' and measure, probabil- preceding if no number is given) the solution and terminology of which are.
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Measure Theory 1.1 : Definition and Introduction
This is the first half of a year-long introduction to probability theory at the graduate level. We will begin by introducing some of the fundamental concepts e. Basic knowledge of abstract measure theory is assumed, but we will endeavor to keep the course as self-contained as possible. We will then discuss the notion of independence and move on to derive the laws of large numbers and some related results. We will conclude with a look at stopping times and random walk. We will follow Durrett's book fairly closely, covering most of the material from chapters A running version of the course notes will be updated every Friday.
Prerequisite: MATH or equivalent. Martingales and martingale convergence theorems if not covered in Weak convergence of measures. Characteristic functions: elementary properties, inversion formula, uniqueness and continuity theorems. Lindeberg-Feller Central Limit Theorem, infinitely divisible laws, stable laws. If time permits: Brownian motion and its properties.