Probability Surveys Articles (Project Euclid)
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The latest articles from Probability Surveys on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 13 Dec 2010 09:16 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Moments of Gamma type and the Brownian supremum process area
http://projecteuclid.org/euclid.ps/1272029738
<strong>Svante Janson</strong><p><strong>Source: </strong>Probab. Surveys, Volume 7, 1--52.</p><p><strong>Abstract:</strong><br/>
We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occurring in the study of hashing with linear displacement, and the general results are applied to these variables.
</p>projecteuclid.org/euclid.ps/1272029738_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTReviewing alternative characterizations of Meixner processhttp://projecteuclid.org/euclid.ps/1311860831<strong>E. Mazzola</strong>, <strong>P. Muliere</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 127--154.</p><p><strong>Abstract:</strong><br/>
Based on the first author’s recent PhD thesis entitled “Profiling processes of Meixner type”, [50] a review of the main characteristics and characterizations of such particular Lévy processes is extracted, emphasizing the motivations for their introduction in literature as reliable financial models. An insight on orthogonal polynomials is also provided, together with an alternative path for defining the same processes. Also, an attempt of simulation of their trajectories is introduced by means of an original R simulation routine.
</p>projecteuclid.org/euclid.ps/1311860831_Thu, 28 Jul 2011 09:47 EDTThu, 28 Jul 2011 09:47 EDTConformally invariant scaling limits in planar critical percolationhttp://projecteuclid.org/euclid.ps/1319806861<strong>Nike Sun</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 155--209.</p><p><strong>Abstract:</strong><br/>
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov’s theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE κ ), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.
</p>projecteuclid.org/euclid.ps/1319806861_Fri, 28 Oct 2011 09:01 EDTFri, 28 Oct 2011 09:01 EDTFundamentals of Stein’s methodhttp://projecteuclid.org/euclid.ps/1319806862<strong>Nathan Ross</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 210--293.</p><p><strong>Abstract:</strong><br/>
This survey article discusses the main concepts and techniques of Stein’s method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning graduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein’s method literature.
</p>projecteuclid.org/euclid.ps/1319806862_Fri, 28 Oct 2011 09:01 EDTFri, 28 Oct 2011 09:01 EDTRecent progress on the Random Conductance Modelhttp://projecteuclid.org/euclid.ps/1325264815<strong>Marek Biskup</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 294--373.</p><p><strong>Abstract:</strong><br/>
Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.
</p>projecteuclid.org/euclid.ps/1325264815_Fri, 30 Dec 2011 12:06 ESTFri, 30 Dec 2011 12:06 ESTTopics on abelian spin models and related problemshttp://projecteuclid.org/euclid.ps/1325264816<strong>Julien Dubédat</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 374--402.</p><p><strong>Abstract:</strong><br/>
In these notes, we discuss a selection of topics on several models of planar statistical mechanics. We consider the Ising, Potts, and more generally abelian spin models; the discrete Gaussian free field; the random cluster model; and the six-vertex model. Emphasis is put on duality, order, disorder and spinor variables, and on mappings between these models.
</p>projecteuclid.org/euclid.ps/1325264816_Fri, 30 Dec 2011 12:06 ESTFri, 30 Dec 2011 12:06 ESTThree theorems in discrete random geometryhttp://projecteuclid.org/euclid.ps/1325264817<strong>Geoffrey Grimmett</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 403--441.</p><p><strong>Abstract:</strong><br/>
These notes are focused on three recent results in discrete random geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is $\sqrt{2+\sqrt{2}}$ ; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on ℤ 2 is $\sqrt{q}/(1+\sqrt{q})$ . Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs.
</p>projecteuclid.org/euclid.ps/1325264817_Fri, 30 Dec 2011 12:06 ESTFri, 30 Dec 2011 12:06 ESTScaling limits and the Schramm-Loewner evolutionhttp://projecteuclid.org/euclid.ps/1325264818<strong>Gregory F. Lawler</strong><p><strong>Source: </strong>Probab. Surveys, Volume 8, 442--495.</p><p><strong>Abstract:</strong><br/>
These notes are from my mini-courses given at the PIMS summer school in 2010 at the University of Washington and at the Cornell probability summer school in 2011. The goal was to give an introduction to the Schramm-Loewner evolution to graduate students with background in probability. This is not intended to be a comprehensive survey of SLE.
</p>projecteuclid.org/euclid.ps/1325264818_Fri, 30 Dec 2011 12:06 ESTFri, 30 Dec 2011 12:06 ESTAround the circular lawhttp://projecteuclid.org/euclid.ps/1325604980<strong>Charles Bordenave</strong>, <strong>Djalil Chafaï</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 1--89.</p><p><strong>Abstract:</strong><br/>
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a n × n random matrix with i.i.d. entries of variance 1/ n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.
</p>projecteuclid.org/euclid.ps/1325604980_Tue, 03 Jan 2012 10:36 ESTTue, 03 Jan 2012 10:36 ESTA lecture on the averaging processhttp://projecteuclid.org/euclid.ps/1327328305<strong>David Aldous</strong>, <strong>Daniel Lanoue</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 90--102.</p><p><strong>Abstract:</strong><br/>
To interpret interacting particle system style models as social dynamics, suppose each pair { i , j } of individuals in a finite population meet at random times of arbitrary specified rates ν ij , and update their states according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values X i ( t − ), X j ( t − ) are replaced by ½( X i ( t − )+ X j ( t − )),½( X i ( t − )+ X j ( t − )). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.
</p>projecteuclid.org/euclid.ps/1327328305_Mon, 23 Jan 2012 09:18 ESTMon, 23 Jan 2012 09:18 ESTSimply generated trees, conditioned Galton–Watson trees, random allocations and condensationhttp://projecteuclid.org/euclid.ps/1331216239<strong>Svante Janson</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 103--252.</p><p><strong>Abstract:</strong><br/>
We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree.
The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model.
This survey paper contains many known results from many different sources, together with some new results.
</p>projecteuclid.org/euclid.ps/1331216239_Thu, 08 Mar 2012 09:17 ESTThu, 08 Mar 2012 09:17 ESTOn temporally completely monotone functions for Markov processeshttp://projecteuclid.org/euclid.ps/1336658304<strong>Francis Hirsch</strong>, <strong>Marc Yor</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 253--286.</p><p><strong>Abstract:</strong><br/>
Any negative moment of an increasing Lamperti process( Y t , t ≥ 0) is a completely monotone function of t . This property enticed us to study systematically, for a given Markov process ( Y t , t ≥ 0), the functions f such that the expectation of f ( Y t ) is a completely monotone function of t . We call these functions temporally completely monotone (for Y ). Our description of these functions is deduced from the analysis made by Ben Saad and Janßen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a Lévy process, a Bessel process, or an increasing Lamperti process.
</p>projecteuclid.org/euclid.ps/1336658304_Thu, 10 May 2012 09:58 EDTThu, 10 May 2012 09:58 EDTSzegö’s theorem and its probabilistic descendantshttp://projecteuclid.org/euclid.ps/1343047754<strong>N.H. Bingham</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 287--324.</p><p><strong>Abstract:</strong><br/>
The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szegö’s work of 1915-21, and has been given a great impetus by the recent work of Simon, in particular his survey paper and three recent books; we allude to the title of the third of these, Szegö’s theorem and its descendants , in ours. Simon’s motivation comes from spectral theory and analysis. Another major area of application of OPUC comes from probability, statistics, time series and prediction theory; see for instance the classic book by Grenander and Szegö, Toeplitz forms and their applications . Coming to the subject from this background, our aim here is to complement this recent work by giving some probabilistically motivated results. We also advocate a new definition of long-range dependence.
</p>projecteuclid.org/euclid.ps/1343047754_Mon, 23 Jul 2012 08:49 EDTMon, 23 Jul 2012 08:49 EDTMultivariate prediction and matrix Szegö theoryhttp://projecteuclid.org/euclid.ps/1343047755<strong>N.H. Bingham</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 325--339.</p><p><strong>Abstract:</strong><br/>
Following the recent survey by the same author of Szegö’s theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).
</p>projecteuclid.org/euclid.ps/1343047755_Mon, 23 Jul 2012 08:49 EDTMon, 23 Jul 2012 08:49 EDTQuasi-stationary distributions and population processeshttp://projecteuclid.org/euclid.ps/1350047379<strong>Sylvie Méléard</strong>, <strong>Denis Villemonais</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 340--410.</p><p><strong>Abstract:</strong><br/>
This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.
</p>projecteuclid.org/euclid.ps/1350047379_Fri, 12 Oct 2012 09:09 EDTFri, 12 Oct 2012 09:09 EDTBougerol’s identity in law and extensionshttp://projecteuclid.org/euclid.ps/1352385533<strong>Stavros Vakeroudis</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 411--437.</p><p><strong>Abstract:</strong><br/>
We present a list of equivalent expressions and extensions of Bougerol’s celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.
</p>projecteuclid.org/euclid.ps/1352385533_Thu, 08 Nov 2012 09:38 ESTThu, 08 Nov 2012 09:38 ESTErratum: Three theorems in discrete random geometryhttp://projecteuclid.org/euclid.ps/1352903647<strong>Geoffrey Grimmett</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 438--438.</p><p><strong>Abstract:</strong><br/>
An error is identified and corrected in the survey entitled ‘Three theorems in discrete random geometry’, published in Probability Surveys 8 (2011) 403–441.
</p>projecteuclid.org/euclid.ps/1352903647_Wed, 14 Nov 2012 09:34 ESTWed, 14 Nov 2012 09:34 ESTQuantile coupling inequalities and their applicationshttp://projecteuclid.org/euclid.ps/1354125785<strong>David M. Mason</strong>, <strong>Harrison H. Zhou</strong><p><strong>Source: </strong>Probab. Surveys, Volume 9, 439--479.</p><p><strong>Abstract:</strong><br/>
This is partly an expository paper. We prove and highlight a quantile inequality that is implicit in the fundamental paper by Komlós, Major, and Tusnády [31] on Brownian motion strong approximations to partial sums of independent and identically distributed random variables. We also derive a number of refinements of this inequality, which hold when more assumptions are added. A number of examples are detailed that will likely be of separate interest. We especially call attention to applications to the asymptotic equivalence theory of nonparametric statistical models and nonparametric function estimation.
</p>projecteuclid.org/euclid.ps/1354125785_Wed, 28 Nov 2012 13:03 ESTWed, 28 Nov 2012 13:03 ESTPlanar percolation with a glimpse of Schramm–Loewner evolutionhttp://projecteuclid.org/euclid.ps/1379686423<strong>Vincent Beffara</strong>, <strong>Hugo Duminil-Copin</strong><p><strong>Source: </strong>Probab. Surveys, Volume 10, 1--50.</p><p><strong>Abstract:</strong><br/>
In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy–Smirnov formula. This theorem, together with the introduction of Schramm–Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $\theta(p)$ for site percolation on the triangular lattice behaves like $(p-p_{c})^{5/36+o(1)}$ as $p\searrow p_{c}=1/2$.
</p>projecteuclid.org/euclid.ps/1379686423_Fri, 20 Sep 2013 10:13 EDTFri, 20 Sep 2013 10:13 EDTOn spectral methods for variance based sensitivity analysishttp://projecteuclid.org/euclid.ps/1385129850<strong>Alen Alexanderian</strong><p><strong>Source: </strong>Probab. Surveys, Volume 10, 51--68.</p><p><strong>Abstract:</strong><br/>
Consider a mathematical model with a finite number of random parameters. Variance based sensitivity analysis provides a framework to characterize the contribution of the individual parameters to the total variance of the model response. We consider the spectral methods for variance based sensitivity analysis which utilize representations of square integrable random variables in a generalized polynomial chaos basis. Taking a measure theoretic point of view, we provide a rigorous and at the same time intuitive perspective on the spectral methods for variance based sensitivity analysis. Moreover, we discuss approximation errors incurred by fixing inessential random parameters, when approximating functions with generalized polynomial chaos expansions.
</p>projecteuclid.org/euclid.ps/1385129850_Fri, 22 Nov 2013 09:17 ESTFri, 22 Nov 2013 09:17 ESTSelf-normalized limit theorems: A surveyhttp://projecteuclid.org/euclid.ps/1385665279<strong>Qi-Man Shao</strong>, <strong>Qiying Wang</strong><p><strong>Source: </strong>Probab. Surveys, Volume 10, 69--93.</p><p><strong>Abstract:</strong><br/>
Let $X_{1},X_{2},\ldots,$ be independent random variables with $EX_{i}=0$ and write $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum $S_{n}/V_{n}$. Other self-normalized limit theorems are also briefly discussed.
</p>projecteuclid.org/euclid.ps/1385665279_Thu, 28 Nov 2013 14:01 ESTThu, 28 Nov 2013 14:01 ESTIntegrable probability: From representation theory to Macdonald processeshttp://projecteuclid.org/euclid.ps/1395076923<strong>Alexei Borodin</strong>, <strong>Leonid Petrov</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, , 1--58.</p><p><strong>Abstract:</strong><br/>
These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.
</p>projecteuclid.org/euclid.ps/1395076923_20140317132205Mon, 17 Mar 2014 13:22 EDTOn the notion(s) of duality for Markov processeshttp://projecteuclid.org/euclid.ps/1398778562<strong>Sabine Jansen</strong>, <strong>Noemi Kurt</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 59--120.</p><p><strong>Abstract:</strong><br/>
We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.
</p>projecteuclid.org/euclid.ps/1398778562_20140429093605Tue, 29 Apr 2014 09:36 EDTStatistical properties of zeta functions’ zeroshttp://projecteuclid.org/euclid.ps/1404393850<strong>Vladislav Kargin</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 121--160.</p><p><strong>Abstract:</strong><br/>
The paper reviews existing results about the statistical distribution of zeros for three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of the main results.
</p>projecteuclid.org/euclid.ps/1404393850_20140703092413Thu, 03 Jul 2014 09:24 EDTCharacterizations of GIG laws: A surveyhttp://projecteuclid.org/euclid.ps/1405603141<strong>Angelo Efoévi Koudou</strong>, <strong>Christophe Ley</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 161--176.</p><p><strong>Abstract:</strong><br/>
Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.
</p>projecteuclid.org/euclid.ps/1405603141_20140717091904Thu, 17 Jul 2014 09:19 EDTDistribution of the sum-of-digits function of random integers: A surveyhttp://projecteuclid.org/euclid.ps/1412947841<strong>Louis H. Y. Chen</strong>, <strong>Hsien-Kuei Hwang</strong>, <strong>Vytas Zacharovas</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 177--236.</p><p><strong>Abstract:</strong><br/>
We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein’s method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.
</p>projecteuclid.org/euclid.ps/1412947841_20141010093046Fri, 10 Oct 2014 09:30 EDTReciprocal processes. A measure-theoretical point of viewhttp://projecteuclid.org/euclid.ps/1412947842<strong>Christian Léonard</strong>, <strong>Sylvie Rœlly</strong>, <strong>Jean-Claude Zambrini</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 237--269.</p><p><strong>Abstract:</strong><br/>
The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal process .
The structures of Markov and reciprocal processes are recalled with emphasis on their time-symmetries. A review of the main properties of the reciprocal processes is presented. Our measure-theoretical approach allows for a unified treatment of the diffusion and jump processes. Abstract results are illustrated by several examples and counter-examples.
</p>projecteuclid.org/euclid.ps/1412947842_20141010093046Fri, 10 Oct 2014 09:30 EDTRegularly varying measures on metric spaces: Hidden regular variation and hidden jumpshttp://projecteuclid.org/euclid.ps/1413896892<strong>Filip Lindskog</strong>, <strong>Sidney I. Resnick</strong>, <strong>Joyjit Roy</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 270--314.</p><p><strong>Abstract:</strong><br/>
We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in [15,24]. Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular-variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in $\mathbb{R}_{+}^{\infty}$ with marginal distributions having regularly varying tails and to càdlàg Lévy processes whose Lévy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.
</p>projecteuclid.org/euclid.ps/1413896892_20141021090816Tue, 21 Oct 2014 09:08 EDTGaussian multiplicative chaos and applications: A reviewhttp://projecteuclid.org/euclid.ps/1415023603<strong>Rémi Rhodes</strong>, <strong>Vincent Vargas</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 315--392.</p><p><strong>Abstract:</strong><br/>
In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are nowadays under active investigation, like the construction of the Liouville measure in $2d$-Liouville quantum gravity or thick points of the Gaussian Free Field. Also, we mention important extensions and generalizations of this theory that have emerged ever since and discuss a whole family of applications, ranging from finance, through the Kolmogorov-Obukhov model of turbulence to $2d$-Liouville quantum gravity. This review also includes new results like the convergence of discretized Liouville measures on isoradial graphs (thus including the triangle and square lattices) towards the continuous Liouville measures (in the subcritical and critical case) or multifractal analysis of the measures in all dimensions.
</p>projecteuclid.org/euclid.ps/1415023603_20141103090646Mon, 03 Nov 2014 09:06 ESTThe trace problem for Toeplitz matrices and operators and its impact in probabilityhttp://projecteuclid.org/euclid.ps/1417528609<strong>Mamikon S. Ginovyan</strong>, <strong>Artur A. Sahakyan</strong>, <strong>Murad S. Taqqu</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 11, 393--440.</p><p><strong>Abstract:</strong><br/>
The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szegö, Toeplitz forms and their applications (University of California Press, Berkeley, 1958). It has then been extensively studied in the literature.
In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes.
The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, hypotheses testing about the spectrum, etc.
We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory.
</p>projecteuclid.org/euclid.ps/1417528609_20141202085652Tue, 02 Dec 2014 08:56 ESTAround Tsirelson’s equation, or: The evolution process may not explain everythinghttp://projecteuclid.org/euclid.ps/1437497795<strong>Kouji Yano</strong>, <strong>Marc Yor</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 12, 1--12.</p><p><strong>Abstract:</strong><br/>
We present a synthesis of a number of developments which have been made around the celebrated Tsirelson’s equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $G$, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $G=\mathbb{T}$.
</p>projecteuclid.org/euclid.ps/1437497795_20150721125636Tue, 21 Jul 2015 12:56 EDTCurrent open questions in complete mixabilityhttp://projecteuclid.org/euclid.ps/1440075824<strong>Ruodu Wang</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 12, 13--32.</p><p><strong>Abstract:</strong><br/>
Complete and joint mixability has raised considerable interest in recent few years, in both the theory of distributions with given margins, and applications in discrete optimization and quantitative risk management. We list various open questions in the theory of complete and joint mixability, which are mathematically concrete, and yet accessible to a broad range of researchers without specific background knowledge. In addition to the discussions on open questions, some results contained in this paper are new.
</p>projecteuclid.org/euclid.ps/1440075824_20150820090346Thu, 20 Aug 2015 09:03 EDTInfinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processeshttp://projecteuclid.org/euclid.ps/1440075825<strong>David Applebaum</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 12, 33--54.</p><p><strong>Abstract:</strong><br/>
We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Lévy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.
</p>projecteuclid.org/euclid.ps/1440075825_20150820090346Thu, 20 Aug 2015 09:03 EDTConformal restriction and Brownian motionhttp://projecteuclid.org/euclid.ps/1444653628<strong>Hao Wu</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 12, 55--103.</p><p><strong>Abstract:</strong><br/>
This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014.
We describe and characterize all random subsets $K$ of simply connected domain which satisfy the “conformal restriction” property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set $K$ in the upper half-plane $\mathbb{H}$ connects two fixed boundary points, say 0 and $\infty$, and given that $K$ stays in a simply connected open subset $H$ of $\mathbb{H}$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is any conformal map from $H$ onto $\mathbb{H}$ fixing 0 and $\infty $. In the radial case, the random set $K$ in the upper half-plane $\mathbb{H}$ connects one fixed boundary points, say 0, and one fixed interior point, say $i$, and given that $K$ stays in a simply connected open subset $H$ of $\mathbb{H}$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is the conformal map from $H$ onto $\mathbb{H}$ fixing 0 and $i$.
It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter $\kappa=8/3$ and Poisson point processes of Brownian excursions and Brownian loops.
</p>projecteuclid.org/euclid.ps/1444653628_20151012084031Mon, 12 Oct 2015 08:40 EDTFractional Gaussian fields: A surveyhttp://projecteuclid.org/euclid.ps/1456149586<strong>Asad Lodhia</strong>, <strong>Scott Sheffield</strong>, <strong>Xin Sun</strong>, <strong>Samuel S. Watson</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 13, 1--56.</p><p><strong>Abstract:</strong><br/>
We discuss a family of random fields indexed by a parameter $s\in\mathbb{R} $ which we call the fractional Gaussian fields , given by \[\mathrm{FGF}_{s}(\mathbb{R} ^{d})=(-\Delta)^{-s/2}W, \] where $W$ is a white noise on $\mathbb{R}^{d}$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H=s-d/2$. In one dimension, examples of $\mathrm{FGF}_{s}$ processes include Brownian motion ($s=1$) and fractional Brownian motion ($1/2<s<3/2$). Examples in arbitrary dimension include white noise ($s=0$), the Gaussian free field ($s=1$), the bi-Laplacian Gaussian field ($s=2$), the log-correlated Gaussian field ($s=d/2$), Lévy’s Brownian motion ($s=d/2+1/2$), and multidimensional fractional Brownian motion ($d/2<s<d/2+1$). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines.
We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the $\mathrm{FGF}_{s}$ with $s\in(0,1)$ can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable Lévy process.
</p>projecteuclid.org/euclid.ps/1456149586_20160222085948Mon, 22 Feb 2016 08:59 ESTHyperbolic measures on infinite dimensional spaceshttp://projecteuclid.org/euclid.ps/1465321421<strong>Sergey G. Bobkov</strong>, <strong>James Melbourne</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 13, 57--88.</p><p><strong>Abstract:</strong><br/>
Localization and dilation procedures are discussed for infinite dimensional $\alpha$-concave measures on abstract locally convex spaces (following Borell’s hierarchy of hyperbolic measures).
</p>projecteuclid.org/euclid.ps/1465321421_20160607134342Tue, 07 Jun 2016 13:43 EDTOn moment sequences and mixed Poisson distributionshttp://projecteuclid.org/euclid.ps/1474374818<strong>Markus Kuba</strong>, <strong>Alois Panholzer</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 13, 89--155.</p><p><strong>Abstract:</strong><br/>
In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable $X$ with moment sequence $(\mu_{s})_{s\in\mathbb{N}}$ we determine a discrete random variable $Y$, whose moment sequence is given by the Stirling transform of the sequence $(\mu_{s})_{s\in\mathbb{N}}$, and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a new simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time $n$. We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics.
</p>projecteuclid.org/euclid.ps/1474374818_20160920083344Tue, 20 Sep 2016 08:33 EDTFrom extreme values of i.i.d. random fields to extreme eigenvalues of finite-volume Anderson Hamiltonianhttp://projecteuclid.org/euclid.ps/1476369041<strong>Arvydas Astrauskas</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 13, 156--244.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to study asymptotic geometric properties almost surely or/and in probability of extreme order statistics of an i.i.d. random field (potential) indexed by sites of multidimensional lattice cube, the volume of which unboundedly increases. We discuss the following topics: (I) high level exceedances, in particular, clustering of exceedances; (II) decay rate of spacings in comparison with increasing rate of extreme order statistics; (III) minimum of spacings of successive order statistics; (IV) asymptotic behavior of values neighboring to extremes and so on. The conditions of the results are formulated in terms of regular variation (RV) of the cumulative hazard function and its inverse. A relationship between RV classes of the present paper as well as their links to the well-known RV classes (including domains of attraction of max-stable distributions) are discussed.
The asymptotic behavior of functionals (I)–(IV) determines the asymptotic structure of the top eigenvalues and the corresponding eigenfunctions of the large-volume discrete Schrödinger operators with an i.i.d. potential (Anderson Hamiltonian). Thus, another aim of the present paper is to review and comment a recent progress on the extreme value theory for eigenvalues of random Schrödinger operators as well as to provide a clear and rigorous understanding of the relationship between the top eigenvalues and extreme values of i.i.d. random potentials. We also discuss their links to the long-time intermittent behavior of the parabolic problems associated with the Anderson Hamiltonian via spectral representation of solutions.
</p>projecteuclid.org/euclid.ps/1476369041_20161013103045Thu, 13 Oct 2016 10:30 EDTStein’s method for comparison of univariate distributionshttp://projecteuclid.org/euclid.ps/1483952471<strong>Christophe Ley</strong>, <strong>Gesine Reinert</strong>, <strong>Yvik Swan</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 14, 1--52.</p><p><strong>Abstract:</strong><br/>
We propose a new general version of Stein’s method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution which is based on a linear difference or differential-type operator. The resulting Stein identity highlights the unifying theme behind the literature on Stein’s method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions: normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Fréchet and Gumbel.
</p>projecteuclid.org/euclid.ps/1483952471_20170109040125Mon, 09 Jan 2017 04:01 ESTFringe trees, Crump–Mode–Jagers branching processes and $m$-ary search treeshttp://projecteuclid.org/euclid.ps/1490169611<strong>Cecilia Holmgren</strong>, <strong>Svante Janson</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 14, 53--154.</p><p><strong>Abstract:</strong><br/>
This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump–Mode–Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees.
We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of $m$-ary search trees in detail; this seems to be new.
Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for $m$-ary search trees, and give for example new results on protected nodes in $m$-ary search trees.
A separate section surveys results on the height of the random trees due to Devroye (1986), Biggins (1995, 1997) and others.
This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees.
</p>projecteuclid.org/euclid.ps/1490169611_20170322040026Wed, 22 Mar 2017 04:00 EDTOpinion exchange dynamicshttp://projecteuclid.org/euclid.ps/1498528815<strong>Elchanan Mossel</strong>, <strong>Omer Tamuz</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 14, 155--204.</p>projecteuclid.org/euclid.ps/1498528815_20170626220021Mon, 26 Jun 2017 22:00 EDTCoagulation and diffusion: A probabilistic perspective on the Smoluchowski PDEhttps://projecteuclid.org/euclid.ps/1512615628<strong>Alan Hammond</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 14, 205--288.</p><p><strong>Abstract:</strong><br/>
The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey’s principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation.
</p>projecteuclid.org/euclid.ps/1512615628_20171206220036Wed, 06 Dec 2017 22:00 ESTCopulas and long memoryhttps://projecteuclid.org/euclid.ps/1513069215<strong>Rustam Ibragimov</strong>, <strong>George Lentzas</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 14, 289--327.</p><p><strong>Abstract:</strong><br/>
This paper focuses on the analysis of persistence properties of copula-based time series. We obtain theoretical results that demonstrate that Gaussian and Eyraud-Farlie-Gumbel-Morgenstern copulas always produce short memory stationary Markov processes. We further show via simulations that, in finite samples, stationary Markov processes, such as those generated by Clayton copulas, may exhibit a spurious long memory-like behavior on the level of copulas, as indicated by standard methods of inference and estimation for long memory time series. We also discuss applications of copula-based Markov processes to volatility modeling and the analysis of nonlinear dependence properties of returns in real financial markets that provide attractive generalizations of GARCH models. Among other conclusions, the results in the paper indicate non-robustness of the copula-level analogues of standard procedures for detecting long memory on the level of copulas and emphasize the necessity of developing alternative inference methods.
</p>projecteuclid.org/euclid.ps/1513069215_20171212040020Tue, 12 Dec 2017 04:00 ESTTASEP hydrodynamics using microscopic characteristicshttps://projecteuclid.org/euclid.ps/1519722018<strong>Pablo A. Ferrari</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 15, 1--27.</p><p><strong>Abstract:</strong><br/>
The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. Along the way we show laws of large numbers for tagged particles, fluxes and second class particles, and simplify existing proofs in the shock cases. The presentation is self contained.
</p>projecteuclid.org/euclid.ps/1519722018_20180227040020Tue, 27 Feb 2018 04:00 ESTTopics in loop measures and the loop-erased walkhttps://projecteuclid.org/euclid.ps/1520326908<strong>Gregory F. Lawler</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 15, 28--101.</p>projecteuclid.org/euclid.ps/1520326908_20180306040150Tue, 06 Mar 2018 04:01 ESTThe Bethe ansatz for the six-vertex and XXZ models: An expositionhttps://projecteuclid.org/euclid.ps/1521079210<strong>Hugo Duminil-Copin</strong>, <strong>Maxime Gagnebin</strong>, <strong>Matan Harel</strong>, <strong>Ioan Manolescu</strong>, <strong>Vincent Tassion</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 15, 102--130.</p><p><strong>Abstract:</strong><br/>
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where $V$ is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a=b=1$ and $c>0$. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly.
Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on $\mathbb{Z}^{2}$ with cluster weight $q>4$ exhibits a first-order phase transition.
</p>projecteuclid.org/euclid.ps/1521079210_20180314220013Wed, 14 Mar 2018 22:00 EDTEquidistribution, uniform distribution: a probabilist’s perspectivehttps://projecteuclid.org/euclid.ps/1524556822<strong>Vlada Limic</strong>, <strong>Nedžad Limić</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 15, 131--155.</p><p><strong>Abstract:</strong><br/>
The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed $\mathsf{mod}~1$ sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original $p$-multiply equidistributed sequence $k^{p}\,t\ \mathsf{mod}~1$, $k\geq1$ (where $p\in\mathbb{N}$ and $t\in[0,1]$), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations.
The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to $\infty$-distributed Koksma’s numbers $t^{k}\ \mathsf{mod}~1$, $k\geq1$ (where $t\in[1,a]$ for some $a>1$), and an important generalization by Niederreiter and Tichy from 1985 are discussed.
The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as certain computer scientists and number theorists.
</p>projecteuclid.org/euclid.ps/1524556822_20180424040026Tue, 24 Apr 2018 04:00 EDTOn the scaling limits of weakly asymmetric bridgeshttps://projecteuclid.org/euclid.ps/1537408916<strong>Cyril Labbé</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 15, 156--242.</p><p><strong>Abstract:</strong><br/>
We consider a discrete bridge from $(0,0)$ to $(2N,0)$ evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order $N^{-\alpha}$ with $\alpha\in(0,\infty)$. We provide a classification of the asymptotic behaviours - invariant measure, hydrodynamic limit and fluctuations - of this model according to the value of the parameter $\alpha$.
</p>projecteuclid.org/euclid.ps/1537408916_20180919220210Wed, 19 Sep 2018 22:02 EDTSandpile modelshttps://projecteuclid.org/euclid.ps/1537776024<strong>Antal A. Járai</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 15, 243--306.</p><p><strong>Abstract:</strong><br/>
This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar’s method. The main ideas of Priezzhev’s computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations.
</p>projecteuclid.org/euclid.ps/1537776024_20180924040037Mon, 24 Sep 2018 04:00 EDTSize bias for one and allhttps://projecteuclid.org/euclid.ps/1546657438<strong>Richard Arratia</strong>, <strong>Larry Goldstein</strong>, <strong>Fred Kochman</strong>. <p><strong>Source: </strong>Probability Surveys, Volume 16, 1--61.</p><p><strong>Abstract:</strong><br/>
Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein’s method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, branching processes, and number theory. In this paper we review the basics and survey some of these unexpected connections.
</p>projecteuclid.org/euclid.ps/1546657438_20190104220359Fri, 04 Jan 2019 22:03 EST