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Über dieses Buch

This book offers a detailed review of perturbed random walks, perpetuities, and random processes with immigration. Being of major importance in modern probability theory, both theoretical and applied, these objects have been used to model various phenomena in the natural sciences as well as in insurance and finance. The book also presents the many significant results and efficient techniques and methods that have been worked out in the last decade.

The first chapter is devoted to perturbed random walks and discusses their asymptotic behavior and various functionals pertaining to them, including supremum and first-passage time. The second chapter examines perpetuities, presenting results on continuity of their distributions and the existence of moments, as well as weak convergence of divergent perpetuities. Focusing on random processes with immigration, the third chapter investigates the existence of moments, describes long-time behavior and discusses limit theorems, both with and without scaling. Chapters four and five address branching random walks and the Bernoulli sieve, respectively, and their connection to the results of the previous chapters.

With many motivating examples, this book appeals to both theoretical and applied probabilists.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Perturbed Random Walks

Abstract
Let \((\xi _{k},\eta _{k})_{k\in \mathbb{N}}\) be a sequence of i.i.d. two-dimensional random vectors with generic copy (ξ, η). No condition is imposed on the dependence structure between ξ and η.
Alexander Iksanov

Chapter 2. Perpetuities

Abstract
Let \((M_{k},Q_{k})_{k\in \mathbb{N}}\) be independent copies of an \(\mathbb{R}^{2}\)-valued random vector (M, Q) with arbitrary dependence of the components, and let X 0 be a random variable which is independent of \((M_{k},Q_{k})_{k\in \mathbb{N}}\).
Alexander Iksanov

Chapter 3. Random Processes with Immigration

Abstract
Denote by \(D(\mathbb{R})\) the Skorokhod space of real-valued right-continuous functions which are defined on \(\mathbb{R}\) and have finite limits from the left at each point of the domain.
Alexander Iksanov

Chapter 4. Application to Branching Random Walk

Abstract
The purpose of this chapter is two-fold. First, we obtain a criterion for uniform integrability of intrinsic martingales \((W_{n})_{n\in \mathbb{N}_{0}}\) in the branching random walk as a corollary to Theorem 2.​1.​1 that provides a criterion for the a.s. finiteness of perpetuities. Second, we state a criterion for the existence of logarithmic moments of a.s. limits of \((W_{n})_{n\in \mathbb{N}_{0}}\) as a corollary to Theorems 1.​3.​1 and 2.​1.​4 While the former gives a criterion for the existence of power-like moments for suprema of perturbed random walks, the latter contains a criterion for the existence of logarithmic moments of perpetuities. To implement the task, we shall exhibit an interesting connection between these at first glance unrelated models which emerges when studying the weighted random tree associated with the branching random walk under the so-called size-biased measure.
Alexander Iksanov

Chapter 5. Application to the Bernoulli Sieve

Abstract
The definition of the Bernoulli sieve which is an infinite allocation scheme can be found on p. 1. Assuming that the number of balls to be allocated equals n (in other words, using a sample of size n from a uniform distribution on [0, 1]), denote by K n the number of occupied boxes and by M n the index of the last occupied box. Also, put L n : = M n K n and note that L n equals the number of empty boxes within the occupancy range (i.e., we only count the empty boxes with indices not exceeding M n ).
Alexander Iksanov

Chapter 6. Appendix

Abstract
A positive measurable function , defined on some neighborhood of , is called slowly varying at if lim t →  ((ut)∕(t)) = 1 for all u > 0.
Alexander Iksanov

Backmatter

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