Topics in Applied Probability:Tempered Stable Distributions, Ad Placement, and Turing's Formula. Lecture 3

Lecture 3, Part 1: Does value-at-risk encourage diversification when losses follow a tempered stable or more general Levy processes? Abstract: Value-at-risk (VaR) is one of the most popular measures of financial risk even though it is not a coherent risk measure and may discourage diversification. In this lecture we show that portfolio selection based on VaR always encourages diversification when losses follow a certain very large class of tempered stable Levy processes. Further, we also give sufficient conditions for more general Levy processes.
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Lecture 3, Part 2: How do tempered stable distributions appear in applications? Abstract: In this lecture we discuss limits theorems for two models of summation of iid random variables with a parameter. The parameter determines how heavy the tails of the distribution are. The first model explains how tempered stable distributions arise in applications. Our discussion is motivated by the problem of modeling mobility, which is important in applications to computer science, anthropology, and ecology.