Level-Crossing Problems and Inverse Gaussian Distributions

1. Introduction: Level-Crossing Problem and Related Fields. 1.1. Sums of independent Random Variables, Gaussian and Inverse Gaussian Distributions. 1.2. Random Walks and Renewal Processes. 1.3. Level-Crossing by a Compound Renewal Process. 1.4. Closed-form Results and Limit Theorems in Level-Crossing. 1.5. Message, Agenda, and Target Audience. 2. Inverse Gaussian and Generalized Inverse Gaussian Distributions. 2.1. Inverse Gaussian Distribution. 2.2. Generalized Inverse Gaussian Distribution. 3. Integral Expressions. 3.1. Elementary Integral Expressions. 3.2. Core Integral Expression. 3.3. Composite Integral Expressions. 4. Distribution of Compound Renewal Process at a Fixed Time Point. 4.1. Compound Renewal Process in Continuous Time. 4.2. Closed-form Results. 4.3. Aspects of Renewal Theory. 4.4. Origin of The Method based on Limit Theorems for Sums. 4.5. Approximation for the Mean. 4.6. Approximation for the Distribution. 4.7. Extensions from Renewal to more General Models. 5. Closed-form Results for the Distribution of First Level-Crossing Time. 5.1. Representations of the Distribution of First Level-Crossing Time. 5.2. Closed-form Results in Exponential Case. 5.3. Closed-form Expression for Conditional Probability. 5.4. Type II Formula and Random Walk with Random Displacements. 5.5. Closed-form Results, When T Is Non-Exponentially Distributed. 5.6. A Result, When Y Is Mixed Exponential. 6. The Inverse Gaussian Approximation. 6.1. Agenda for This Chapter. 6.2. Statement of Main Results. 6.3. Shorthand Notation, Structure Lemmas, Identities Specific to a choice of Arguments, and Centering At Z = 0. 6.4. Expressions of The First Kind. 6.5. Expressions of The Second Kind. 6.6. Expressions of the third kind 6.7. Proof of Theorem 6.1. 6.8. Numerical Illustrations. 6.9. Conclusions. 7. Refinement of the Inverse Gaussian Approximation. 7.1 Asymptotic Expansions: Rigorous Vs. Heuristic. 7.2. Expansion for the Distribution of the First Level-Crossing Time. 7.3. Proof of Theorem 7.1. 7.4. Numerical Illustrations. 8. Derivatives of The First Level-Crossing Time Distribution. 8.1. The Problem and Its Rationale. 8.2. Approximations for Derivatives. 8.3. Fundamental Identities for Derivatives with Respect to C and U. 8.4. Proof of Theorem 8.1. 8.5. Proof of Theorem 8.2. 9. A Breakthrough in the Level-Crossing Problem. 9.1. Neyman's Cycles in Level-Crossing Problem. 9.2. Normal Approximation Versus Inverse Gaussian Approximation. 9.3. Diffusion Approximation Versus inverse Gaussian Approximation: Is there A Mix-Up in This Collation? 9.4. Teugels' Approximation Versus Inverse Gaussian Approximation. 9.5. Conclusions. Appendices. Index.