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2021 | Book

A Lifetime of Excursions Through Random Walks and Lévy Processes Read chapter Read first chapter

A Lifetime of Excursions Through Random Walks and Lévy Processes

A Volume in Honour of Ron Doney’s 80th Birthday

Editors: Loïc Chaumont, Andreas E. Kyprianou

Publisher: Springer International Publishing

Book Series : Progress in Probability

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
A Lifetime of Excursions Through Random Walks and Lévy Processes
Abstract
We recall the many highlights of Professor Ron Doney’s career summarising his main contributions to the theory of random walks and Lévy processes.
Loïc Chaumont, Andreas E. Kyprianou
Path Decompositions of Perturbed Reflecting Brownian Motions
Abstract
We are interested in path decompositions of a perturbed reflecting Brownian motion (PRBM) at the hitting times and at the minimum. Our study relies on the loop soups developed by Lawler and Werner (Probab Theory Relat Fields 4:197–217, 2004) and Le Jan (Ann Probab 38:1280–1319, 2010; Markov Paths, Loops and Fields. École d’été Saint-Flour XXXVIII 2008. Lecture Notes in Mathematics vol 2026. Springer, Berlin, 2011), in particular on a result discovered by Lupu (Mém Soc Math Fr (N.S.) 158, 2018) identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.
Elie Aïdékon, Yueyun Hu, Zhan Shi
On Doney’s Striking Factorization of the Arc-Sine Law
Abstract
In Doney (Bull Lond Math Soc 19(2):177–182, 1987), R. Doney identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.
Larbi Alili, Carine Bartholmé, Loïc Chaumont, Pierre Patie, Mladen Savov, Stavros Vakeroudis
On a Two-Parameter Yule-Simon Distribution
Abstract
We extend the classical one-parameter Yule-Simon law to a version depending on two parameters, which in part appeared in Bertoin (J Stat Phys 176(3):679–691, 2019) in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in Bertoin (J Stat Phys 176(3):679–691, 2019). Finally, by superposing mutations to the branching process, we propose a model which leads to the two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon (Biometrika 42(3/4):425–440, 1955) on limiting word frequencies.
Erich Baur, Jean Bertoin
The Limit Distribution of a Singular Sequence of Itô Integrals
Abstract
We give an alternative, elementary proof of a result of Peccati and Yor concerning the limit law of a sequence of Itô integrals with integrands having singular asymptotic behavior.
Denis Bell
On Multivariate Quasi-infinitely Divisible Distributions
Abstract
A quasi-infinitely divisible distribution on \(\mathbb {R}^d\) is a probability distribution μ on \(\mathbb {R}^d\) whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on \(\mathbb {R}^d\). Equivalently, it can be characterised as a probability distribution whose characteristic function has a Lévy–Khintchine type representation with a “signed Lévy measure”, a so called quasi–Lévy measure, rather than a Lévy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato (Trans Am Math Soc 370:8483–8520, 2018). The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on \(\mathbb {Z}^d\)-valued quasi-infinitely divisible distributions.
David Berger, Merve Kutlu, Alexander Lindner
Extremes and Regular Variation
Abstract
We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the point of view of our recent work on general regular variation.
Nick H. Bingham, Adam J. Ostaszewski
Some New Classes and Techniques in the Theory of Bernstein Functions
Abstract
In this paper we provide some new properties that are complementary to the book of Schilling-Song-Vondraček (Bernstein functions, 2nd edn. De Gruyter, Berlin, 2012).
Safa Bridaa, Sonia Fourati, Wissem Jedidi
A Transformation for Spectrally Negative Lévy Processes and Applications
Abstract
The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, Patie (Bull Sci Math 133(4):355–382, 2009; Bernoulli 17(2):814–826, 2011), Kyprianou and Patie (Ann Inst H Poincar’ Probab Statist 47(3):917–928, 2011), Gnedin (Regeneration in Random Combinatorial Structures. arXiv:0901.4444v1 [math.PR]), Patie and Savov (Electron J Probab 17(38):1–22, 2012), as well as in older work of Urbanik (Probab Math Statist 15:493–513, 1995). We show how some specific instances of this mapping prove to be useful for a variety of applications.
Marie Chazal, Andreas E. Kyprianou, Pierre Patie
First-Passage Times for Random Walks in the Triangular Array Setting
Abstract
In this paper we continue our study of exit times for random walks with independent but not necessarily identically distributed increments. Our paper “First-passage times for random walks with non-identically distributed increments” (2018) was devoted to the case when the random walk is constructed by a fixed sequence of independent random variables which satisfies the classical Lindeberg condition. Now we consider a more general situation when we have a triangular array of independent random variables. Our main assumption is that the entries of every row are uniformly bounded by a deterministic sequence, which tends to zero as the number of the row increases.
Denis Denisov, Alexander Sakhanenko, Vitali Wachtel
On Local Times of Ornstein-Uhlenbeck Processes
Abstract
We establish expressions of the local time process of an Ornstein-Uhlenbeck process in terms of the local times on curves of a Brownian motion.
Nathalie Eisenbaum
Two Continua of Embedded Regenerative Sets
Abstract
Given a two-sided real-valued Lévy process \((X_t)_{t \in \mathbb {R}}\), define processes \((L_t)_{t \in \mathbb {R}}\) and \((M_t)_{t \in \mathbb {R}}\) by \(L_t := \sup \{h \in \mathbb {R} : h - \alpha (t-s) \le X_s \text{ for all } s \le t\} = \inf \{X_s + \alpha (t-s) : s \le t\}\), \(t \in \mathbb {R}\), and \(M_t := \sup \{ h \in \mathbb {R} : h - \alpha |t-s| \leq X_s \text{ for all } s \in \mathbb {R} \} = \inf \{X_s + \alpha |t-s| : s \in \mathbb {R}\}\), \(t \in \mathbb {R}\). The corresponding contact sets are the random sets \(\mathcal {H}_\alpha := \{ t \in \mathbb {R} : X_{t}\wedge X_{t-} = L_t\}\) and \(\mathcal {Z}_\alpha := \{ t \in \mathbb {R} : X_{t}\wedge X_{t-} = M_t\}\). For a fixed \(\alpha >\mathbb {E}[X_1]\) (resp. \(\alpha >|\mathbb {E}[X_1]|\)) the set \(\mathcal {H}_\alpha \) (resp. \(\mathcal {Z}_\alpha \)) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections \((\mathcal {H}_{\alpha })_{\alpha > \mathbb {E}[X_1]}\) and \((\mathcal {Z}_{\alpha })_{\alpha > |\mathbb {E}[X_1]|}\) are increasing in α and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that \((\sup \{t < 0 : t \in \mathcal {H}_\alpha \})_{\alpha > \mathbb {E}[X_1]}\) is a càdlàg, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for \((\sup \{t < 0 : t \in \mathcal {Z}_\alpha \})_{\alpha > |\beta |}\) when \((X_t)_{t \in \mathbb {R}}\) is a (two-sided) Brownian motion with drift β.
Steven N. Evans, Mehdi Ouaki
No-Tie Conditions for Large Values of Extremal Processes
Abstract
We give necessary and sufficient conditions for there to be no ties, asymptotically, among large values of a space-time Poisson point process evolving homogeneously in time. The convergence is at small times, in probability or almost sure.
Yuguang Ipsen, Ross Maller
Slowly Varying Asymptotics for Signed Stochastic Difference Equations
Abstract
For a stochastic difference equation D n = A n D n−1 + B n which stabilises upon time we study tail distribution asymptotics for D n under the assumption that the distribution of \(\log (1+|A_1|+|B_1|)\) is heavy-tailed, that is, all its positive exponential moments are infinite. The aim of the present paper is three-fold. Firstly, we identify the asymptotic behaviour not only of the stationary tail distribution but also of D n. Secondly, we solve the problem in the general setting when A takes both positive and negative values. Thirdly, we get rid of auxiliary conditions like finiteness of higher moments introduced in the literature before.
Dmitry Korshunov
The Doob–McKean Identity for Stable Lévy Processes
Abstract
We re-examine the celebrated Doob–McKean identity that identifies a conditioned one-dimensional Brownian motion as the radial part of a 3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now in the analogous setting of isotropic α-stable processes. We find a natural analogue that matches the Brownian setting, with the role of the Brownian motion replaced by that of the isotropic α-stable process, providing one interprets the components of the original identity in the right way.
Andreas E. Kyprianou, Neil O’Connell
Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes
Abstract
Suppose that S is a closed set of the unit sphere \(\mathbb {S}^{d-1} = \{x\in \mathbb {R}^d: |x| =1\}\) in dimension d ≥ 2, which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension d ≥ 2 conditioned to approach S continuously, allowing for the interior and exterior of \(\mathbb {S}^{d-1}\) to be visited infinitely often. Additionally, we show that this process is in duality with the unconditioned stable Lévy process. We can replicate the aforementioned results by similar ones in the setting that S is replaced by D, a closed bounded subset of the hyperplane \(\{x\in \mathbb {R}^d : (x, v) = 0\}\) with positive surface measure, where v is the unit orthogonal vector and where (⋅, ⋅) is the usual Euclidean inner product. Our results complement similar results of the authors [17] in which the stable process was further constrained to attract to and repel from S from either the exterior or the interior of the unit sphere.
Mateusz Kwaśnicki, Andreas E. Kyprianou, Sandra Palau, Tsogzolmaa Saizmaa
Angular Asymptotics for Random Walks
Abstract
We study the set of directions asymptotically explored by a spatially homogeneous random walk in d-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some examples. We also explore links to the asymptotics of one-dimensional projections, and to the growth of the convex hull of the random walk.
Alejandro López Hernández, Andrew R. Wade
First Passage Times of Subordinators and Urns
Abstract
It is well-known that the first time a stable subordinator reaches [1, +∞). is Mittag-Leffler distributed. These distributions also appear as limiting distributions in triangular Polya urns. We give a direct link between these two results, using a previous construction of the range of stable subordinators. Beyond the stable case, we show that for a subclass of complete subordinators in the domain of attraction of stable subordinators, the law of the first passage time is given by the limit of an urn with the same replacement rule but with a random initial composition.
Philippe Marchal
Metadata
Title
A Lifetime of Excursions Through Random Walks and Lévy Processes
Editors
Loïc Chaumont
Andreas E. Kyprianou
Copyright Year
2021
Electronic ISBN
978-3-030-83309-1
Print ISBN
978-3-030-83308-4
DOI
https://doi.org/10.1007/978-3-030-83309-1