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April 20, 2026, 7:21 a.m.
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| Iksanov A. Locally Perturbed Random Walks 2025.pdf | 3.2 MB |
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SOURCE: Iksanov A. Locally Perturbed Random Walks 2025
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MEDIAINFO
Textbook in PDF format This monograph provides a comprehensive overview of locally perturbed random walks, tools used for their analysis, and current research on their applications. The authors present the material in a self-contained manner, providing strong motivation in Chapter One with illustrative examples of locally perturbed random walks and an introduction of the mathematical tools that are used throughout the book. Chapter Two shows the construction of various stochastic processes that serve as scaling limits for locally perturbed random walks, particularly focusing on reflected and skewed processes. In Chapter Three, the authors prove various limit theorems for these perturbed random walks. The final chapter serves as an appendix that collects essential background material for readers who wish to understand the arguments more deeply. Locally Perturbed Random Walks will appeal to researchers interested in this area within modern probability theory. It is also accessible to students who have taken a second course in probability. Preface List of Notation Introduction Lévy-Type Processes with Singularities Relected Processes The Skorokhod Reflection Problem The Generalized Skorokhod Reflection Problem Properties of a Reflected Brownian Motion with Jump-Type Exit from 0 Bibliographic Comments The Skew Brownian Motion and Its Generalizations The Skew Brownian Motion The Walsh Brownian Motion Bibliographic Comments The Skew Stable Lévy Process Definition Proof of Theorem 2.3.1 An Equation for the Skew Stable Lévy Process Bibliographic Comments Functional Limit Theorems for Locally Perturbed Random Walks Preliminaries Definition of Random Walk with Membrane Examples of Random Walks with Reflection Functional Limit Theorems for Random Walks with Reflection Perturbations with Finite Mean On Two-Stage Models and the Generalized Reflection Perturbations with Regularly Varying Distribution Tails Perturbations with Slowly Varying Distribution Tails Random Walks with Membrane and a Skew Brownian Motion Convergence to a Walsh Brownian Motion Examples Proof of Theorem 3.3.2 Proof of Theorem 3.3.3 Proof Corollary 3.3.1 Limit Theorems for Heavy-Tailed Random Walks with Membrane at 0 Main Result Proof of Theorem 3.4.1(a) Proof of Theorem 3.4.1(b) Multidimensional Random Walks with Membranes Two-Dimensional Random Walks with Finite Membranes Proof of Theorem 3.5.1 Proof of Theorem 3.5.2 Multidimensional Random Walks with Periodic Membranes Concentrated on Hyperplanes Bibliographic Comments Auxiliary Results Probability Measures and Weak Convergence The Space of Continuous Functions C([0,∞),Rd) Weak Convergence and the Locally Uniform Topology The Wiener Measure and Donsker's Invariance Principle The Skorokhod Space D([0,∞),Rd) The J1-Topology Convergence of Probability Measures on D([0,∞),Rd) Continuity of Mappings on D([0,∞),Rd) Generalized Inverse Functions Stable Distributions Domains of Attraction of Stable Distributions Convergence of Markov Processes Itô's Excursion Theory Bibliography Index
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