A Lifetime of Excursions Through Random Walks and Lévy Processes : A Volume in Honour of Ron Doney’s 80th Birthday / / edited by Loïc Chaumont, Andreas E. Kyprianou

• Record Nr.: UNINA9910520083903321
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2021
• ISBN: 3-030-83309-7
• Edizione: [1st ed. 2021.]
• Descrizione fisica: 1 online resource (354 pages)
• Collana: Progress in Probability, , 2297-0428 ; ; 78
• Disciplina: 519.282
• Soggetti: Probabilities; Discrete mathematics; Probability Theory; Discrete Mathematics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Contents -- A Lifetime of Excursions Through Random Walks and Lévy Processes -- References -- Path Decompositions of Perturbed Reflecting Brownian Motions -- 1 Introduction -- 2 Preliminaries -- 2.1 The Brownian Loop Soup -- 2.2 The Poisson-Dirichlet Distribution -- 2.3 Some Known Results -- 3 Decomposition at a Hitting Time -- 4 Decomposition at the Minimum -- 5 The Perturbed Bessel Process and Its Rescaling at a Stopping Time -- References -- On Doney's Striking Factorization of the Arc-Sine Law -- 1 Introduction -- 2 The Arc-Sine Law and Exponential Functional of Lévy Processes -- 3 Proofs -- 3.1 Proof of Corollary 2.2 -- 3.2 Proof of Corollary 2.3 -- 3.3 Proof of Corollary 2.4 -- References -- On a Two-Parameter Yule-Simon Distribution -- 1 Introduction -- 2 Preliminaries on the General Branching Process Yθ -- 3 Poissonian Representation for the Tail Distribution -- 4 Tail Asymptotic Behaviors -- 5 Connection with a Population Model with Neutral Mutations -- 5.1 Simon's Model in Terms of Yule Processes with Mutations -- 5.2 A Generalization of Simon's Model -- The Full Range of the One-Parameter Yule-Simon Law -- A Two-Parameter Generalization -- References -- The Limit Distribution of a Singular Sequence of Itô Integrals -- 1 Introduction ; and Statement of the Theorem -- 2 Proof of the Theorem -- 3 Concluding Remarks -- References -- On Multivariate Quasi-infinitely Divisible Distributions -- 1 Introduction -- 2 The Lévy-Khintchine Type Representation -- 3 Examples -- 4 Conditions for Absolute Continuity -- 5 Topological Properties of the Class of Infinitely Divisible Distributions -- 6 Conditions for Weak Convergence -- 7 Support Properties -- 8 Moments -- References -- Extremes and Regular Variation -- 1 One Dimension -- 2 Higher Dimensions -- 3 Historical Comments -- References.; Some New Classes and Techniques in the Theory of BernsteinFunctions -- 1 A Unified View on Subclasses of Bernstein Functions -- 2 Investigating the Class CBFa -- 3 A New Injective Mapping from BF onto CBF -- References -- A Transformation for Spectrally Negative Lévy Processes and Applications -- 1 Introduction -- 2 The Transformation Tδ,β -- 3 Scale Functions for Spectrally Negative Lévy Processes -- 4 Exponential Functional and Length-Biased Distribution -- 4.1 The Case of Subordinators -- 4.2 The Spectrally Negative Case -- 5 Entrance Laws and Intertwining Relations of pssMp -- References -- First-Passage Times for Random Walks in the Triangular ArraySetting -- 1 Introduction and the Main Result -- 1.1 Introduction -- 1.2 Main Result -- 1.3 Triangular Arrays of Weighted Random Variables -- 1.4 Discussion of the Assumption (13) -- 2 Proofs -- 2.1 Some Estimates in the Central Limit Theorem -- 2.2 Estimates for Expectations of Z"0362Zk -- 2.3 Proof of Theorem 1 -- 2.4 Proof of Corollary 3 -- 2.5 Calculations Related to Example 7 -- References -- On Local Times of Ornstein-Uhlenbeck Processes -- 1 Introduction and Main Result -- 2 Local Times on Curves -- 3 Proof of Theorem 1.1 -- 4 Some Asymptotics of Local Times -- References -- Two Continua of Embedded Regenerative Sets -- 1 Introduction -- 2 Regenerative Sets -- 3 Relationship with the Set of Ladder Times -- 4 Regenerative Embedding Generalities -- 5 A Continuous Family of Embedded Regenerative Sets -- 6 Another Continuous Family of Embedded Regenerative Sets -- 7 Some Real Analysis -- References -- No-Tie Conditions for Large Values of Extremal Processes -- 1 Introduction -- 1.1 Poisson Point Processes -- 1.2 Extremal Processes -- 2 No Ties in Probability -- 3 No Ties Almost Surely -- 4 Sufficient Conditions and an Example -- References.; Slowly Varying Asymptotics for Signed Stochastic DifferenceEquations -- 1 Introduction -- 2 Positive Stochastic Difference Equation -- 3 Impact of Atom at Zero -- 4 The Case of Positive A and Signed B -- 5 Balance of Negative and Positive Tails in the Case of Signed A -- References -- The Doob-McKean Identity for Stable Lévy Processes -- 1 Introduction -- 2 Doob-McKean for Isotropic α-Stable Processes -- 3 The Special Case of Cauchy Processes -- 3.1 Lamperti Representation of the Doob-McKean Identity -- 3.2 Connection to Cauchy Process Conditioned to Stay Positive -- 3.3 Pathwise Representation -- 3.4 Generators -- 4 Concluding Remarks -- References -- Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes -- 1 Introduction -- 2 Oscillatory Attraction Towards S -- 3 Oscillatory Repulsion from S and Duality -- 4 The Setting of a Subset in an Rd-1 Hyperplane -- 5 Heuristic for the Proof of Theorem 2 -- 6 Proof of Theorem 2 (i) -- 7 Proof of Theorem 2 (ii) -- 8 Proof of Theorem 1 -- 9 Proof of Theorem 3 -- 9.1 Completing the Proof of Theorem 3 -- 10 Proof of Theorem 4 -- Appendix: Hypergeometric Identities -- References -- Angular Asymptotics for Random Walks -- 1 Introduction -- 2 Recurrent Directions -- 3 Compactification and Growth Rates -- 4 Limiting Direction -- 5 The Zero-Drift Case -- 6 An Arbitrary Set of Recurrent ; Directions -- 7 Convexity and an Upper Bound -- 8 Projection Asymptotics -- 9 The Convex Hull -- 10 Some Examples -- 11 Concluding Remarks -- A The Recurrent Case -- References -- First Passage Times of Subordinators and Urns -- 1 Introduction -- 2 A Construction of Regenerative Sets -- 2.1 The Lattice Case -- 2.2 The Continuous Case -- 2.3 Relating the Discrete and the Continuous Case -- 3 Embedded Urns -- 3.1 An Alternative Description of the Urn -- 3.2 The Stable Case.; 3.3 The General Case -- 4 Proof of Theorem 1 -- References.
• Sommario/riassunto: This collection honours Ron Doney’s work and includes invited articles by his collaborators and friends. After an introduction reviewing Ron Doney’s mathematical achievements and how they have influenced the field, the contributed papers cover both discrete-time processes, including random walks and variants thereof, and continuous-time processes, including Lévy processes and diffusions. A good number of the articles are focused on classical fluctuation theory and its ramifications, the area for which Ron Doney is best known.