An introduction to the class of regularly varying distributions and the important properties of this class, including scale invariance. Examples of applying regularly varying distributions to branching processes are included.

An introduction to the emergence of heavy-tailed distributions in the context of multiplicative processes.The multiplicative central limit theorem is presented, and variations of multiplicative processes with lower barriers and noise are studied.Further, an example of the emergence of heavy tails in random graphs via preferential attachment is included.

Extremal approaches for semi-parametric estimation of power-law tails are presented, including the Hill estimator, the moments estimate, the Pickands estimator, and Peaks over threshold.Further, approaches for estimating where the tail begins are presented, including PLFIT and the double bootstrap method.

An introduction to the class of long-tailed distributions and the important properties of this class, including properties of the hazard rate and the residual life distribution. Examples applying long-tailed distributions to random extrema are included.

Classical approaches for parametric estimation of power-laws distributions are presented, including (weighted) linear regression and maximum likelihood estimation.

An introduction to the emergence of heavy-tailed distributions in the context of additive processes.Stable distributions are introduced and the generalized central limit theorem is presented.Further, an example of the emergence of heavy tails in random walks is included.

An introduction to the class of subexponential distributions and the important properties of this class, including the catastrophe principle. Examples applying subexponential distributions to random sums and random walks are included.

This chapter analyzes linear and nonlinear discrete-time systems described by a discrete-time state-space model whose inputs are uncertain but known to belong to an ellipsoid. For the linear case, even if the input set is an ellipsoid, the set containing all possible values that the state can take is not an ellipsoid in general, but it can be upper bounded by an ellipsoid. We develop techniques for recursively computing a family of such upper-bounding ellipsoids. Within this family, we then show how to choose ellipsoids that are optimal in some sense, e.g., they have minimum volume. For the nonlinear case, we will again resort to linearization techniques to approximately characterize the set containing all possible values that the state can take. The application of the techniques presented is illustrated using the same inertia-less AC microgrid model used in Chapter 5.

This chapter starts by reviewing important concepts from probability theory and stochastic processes. Subsequent chapters on probabilistic input and structural uncertainty make heavy use of random vectors and vector-valued stochastic process, so the reader should be familiar with the material included on these concepts. Next, the chapter provides a review of set-theoretic notions. The material on sets in Euclidean space included in this part is key to understanding the set-theoretic approach to input uncertainty modeling. The chapter concludes with a review of several fundamental concepts from the theory of discrete- and continuous-time linear dynamical systems.

We prove the converse of Stone’s theorem.