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Kosmos
Astronomia Astrofizyka
Inne

Kultura
Sztuka dawna i współczesna, muzea i kolekcje

Metoda
Metodologia nauk, Matematyka, Filozofia, Miary i wagi, Pomiary

Materia
Substancje, reakcje, energia
Fizyka, chemia i inżynieria materiałowa

Człowiek
Antropologia kulturowa Socjologia Psychologia Zdrowie i medycyna

Wizje
Przewidywania Kosmologia Religie Ideologia Polityka

Ziemia
Geologia, geofizyka, geochemia, środowisko przyrodnicze

Życie
Biologia, biologia molekularna i genetyka

Cyberprzestrzeń
Technologia cyberprzestrzeni, cyberkultura, media i komunikacja

Działalność
Wiadomości | Gospodarka, biznes, zarządzanie, ekonomia

Technologie
Budownictwo, energetyka, transport, wytwarzanie, technologie informacyjne

# Probability Surveys

Czasopismo Open Access z projektu Euclid

## Characterizations of GIG laws: A survey

Angelo Efoévi Koudou, Christophe Ley. Source: Probability Surveys, Volume 11, 161--176.Abstract:

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.

2014/07/17 - 16:58

## Statistical properties of zeta functions’ zeros

Vladislav Kargin. Source: Probability Surveys, Volume 11, 121--160.Abstract:

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.

2014/07/04 - 22:54

## On the notion(s) of duality for Markov processes

Sabine Jansen, Noemi Kurt. Source: Probability Surveys, Volume 11, 59--120.Abstract:
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.

2014/04/30 - 16:39

## Integrable probability: From representation theory to Macdonald processes

Alexei Borodin, Leonid Petrov. Source: Probability Surveys, Volume 11, , 1--58.Abstract:

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.

2014/03/18 - 12:10

## Integrable probability: From representation theory to Macdonald processes

Alexei Borodin, Leonid Petrov. Source: Probability Surveys, Volume 11, , 1--58.Abstract:

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.

2014/03/18 - 12:10

## Self-normalized limit theorems: A survey

Qi-Man Shao, Qiying WangSource: Probab. Surveys, Volume 10, 69--93.Abstract:

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.

2013/11/29 - 14:09

## On spectral methods for variance based sensitivity analysis

Alen AlexanderianSource: Probab. Surveys, Volume 10, 51--68.Abstract:

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.

2013/11/22 - 22:53

## Planar percolation with a glimpse of Schramm–Loewner evolution

Vincent Beffara, Hugo Duminil-CopinSource: Probab. Surveys, Volume 10, 1--50.Abstract:

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 $p\searrow p_{c}=1/2$.

2013/09/21 - 17:00

## Quantile coupling inequalities and their applications

David M. Mason, Harrison H. ZhouSource: Probab. Surveys, Volume 9, 439--479.Abstract:

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.

2012/11/29 - 10:15

## Erratum: Three theorems in discrete random geometry

Geoffrey GrimmettSource: Probab. Surveys, Volume 9, 438--438.Abstract:

An error is identified and corrected in the survey entitled ‘Three theorems in discrete random geometry’, published in Probability Surveys 8 (2011) 403–441.

2012/11/14 - 21:06

## Bougerol’s identity in law and extensions

Stavros VakeroudisSource: Probab. Surveys, Volume 9, 411--437.Abstract:

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.

2012/11/09 - 16:08

## Quasi-stationary distributions and population processes

Sylvie Méléard, Denis VillemonaisSource: Probab. Surveys, Volume 9, 340--410.Abstract:

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.

2012/10/13 - 01:37

## Szegö’s theorem and its probabilistic descendants

N.H. BinghamSource: Probab. Surveys, Volume 9, 287--324.Abstract:

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.

2012/07/23 - 17:39

## Multivariate prediction and matrix Szegö theory

N.H. BinghamSource: Probab. Surveys, Volume 9, 325--339.Abstract:

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).

2012/07/23 - 17:39

## On temporally completely monotone functions for Markov processes

Francis Hirsch, Marc YorSource: Probab. Surveys, Volume 9, 253--286.Abstract:

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.

2012/05/12 - 07:23

## Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation

Svante JansonSource: Probab. Surveys, Volume 9, 103--252.Abstract:

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.

2012/03/09 - 00:34

## A lecture on the averaging process

David Aldous, Daniel LanoueSource: Probab. Surveys, Volume 9, 90--102.Abstract:

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.

2012/01/24 - 09:24

## Around the circular law

Charles Bordenave, Djalil ChafaïSource: Probab. Surveys, Volume 9, 1--89.Abstract:

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.

2012/01/04 - 05:38

## Recent progress on the Random Conductance Model

Marek BiskupSource: Probab. Surveys, Volume 8, 294--373.Abstract:

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.

2011/12/31 - 08:06

## Topics on abelian spin models and related problems

Julien DubédatSource: Probab. Surveys, Volume 8, 374--402.Abstract:

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.

2011/12/31 - 08:06

## Three theorems in discrete random geometry

Geoffrey GrimmettSource: Probab. Surveys, Volume 8, 403--441.Abstract:

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.

2011/12/31 - 08:06

## Scaling limits and the Schramm-Loewner evolution

Gregory F. LawlerSource: Probab. Surveys, Volume 8, 442--495.Abstract:

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.

2011/12/31 - 08:06

## Conformally invariant scaling limits in planar critical percolation

Nike SunSource: Probab. Surveys, Volume 8, 155--209.Abstract:

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.

2011/10/30 - 07:32

## Fundamentals of Stein’s method

Nathan RossSource: Probab. Surveys, Volume 8, 210--293.Abstract:

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.

2011/10/30 - 07:32

## A basic theory of Benford’s Law

Arno Berger, Theodore P. HillSource: Probab. Surveys, Volume 8, 1--126.Abstract:

Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s Law (BL) or, in a special case, as the First Digit Law . The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.

2011/07/29 - 09:17

## Reviewing alternative characterizations of Meixner process

E. Mazzola, P. MuliereSource: Probab. Surveys, Volume 8, 127--154.Abstract:

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.

2011/07/29 - 09:17

## Moments of Gamma type and the Brownian supremum process area

Svante JansonSource: Probab. Surveys, Volume 7, 1--52.Abstract:

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.

2010/12/30 - 03:10

## Limit theorems for discrete-time metapopulation models

F.M. Buckley, P.K. PollettSource: Probab. Surveys, Volume 7, 53--83.Abstract:

We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.

2010/12/30 - 03:10

## Symbolic extensions of smooth interval maps

Tomasz DownarowiczSource: Probab. Surveys, Volume 7, 84--104.Abstract:

In this course we will present the full proof of the fact that every smooth dynamical system on the interval or circle X , constituted by the forward iterates of a function f : X → X which is of class C r with r > 1, admits a symbolic extension, i.e., there exists a bilateral subshift ( Y , S ) with Y a closed shift-invariant subset of Λ ℤ , where Λ is a finite alphabet, and a continuous surjection π : Y → X which intertwines the action of f (on X ) with that of the shift map S (on Y ). Moreover, we give a precise estimate (from above) on the entropy of each invariant measure ν supported by Y in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure μ on X , the “Lyapunov exponent” of μ (the genuine Lyapunov exponent for ergodic μ , otherwise its analog), and the smoothness parameter r . This estimate agrees with a conjecture formulated in [15] around 2003 for smooth dynamical systems on manifolds.

2010/12/30 - 03:10

## Regeneration in random combinatorial structures

Alexander V. GnedinSource: Probab. Surveys, Volume 7, 105--156.Abstract:

Kingman’s theory of partition structures relates, via a natural sampling procedure, finite partitions to hypothetical infinite populations. Explicit formulas for distributions of such partitions are rare, the most notable exception being the Ewens sampling formula, and its two-parameter extension by Pitman. When one adds an extra structure to the partitions like a linear order on the set of blocks and regenerative properties, some representation theorems allow to get more precise information on the distribution. In these notes we survey recent developments of the theory of regenerative partitions and compositions. In particular, we discuss connection between ordered and unordered structures, regenerative properties of the Ewens-Pitman partitions, and asymptotics of the number of components.

2010/12/30 - 03:10

## Combinatorics and cluster expansions

William G. FarisSource: Probab. Surveys, Volume 7, 157--206.Abstract:

This article is about the connection between enumerative combinatorics and equilibrium statistical mechanics. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. The passage from species to generating functions is a combinatorial analog of the Fourier transform. Indeed, there is a convolution multiplication on species that is mapped to a pointwise multiplication of the exponential generating functions. The statistical mechanics side deals with a probability model of an equilibrium gas. The cluster expansion that gives the density of the gas is the exponential generating function for the species of rooted connected graphs. The main results of the theory are simple criteria that guarantee the convergence of this expansion. It turns out that other problems in combinatorics and statistical mechanics can be translated to this gas setting, so it is a universal prescription for dealing with systems of high dimension.

2010/12/30 - 03:10

## Addendum to Moments of Gamma type and the Brownian supremum process area

Svante JansonSource: Probab. Surveys, Volume 7, 207--208.Abstract:

Supplementary references and material are provided to the paper entitled ‘Moments of Gamma type and the Brownian supremum process area’, published in Probability Surveys 7 (2010) 1–52.

References:[1] B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals. Compositio Math. 15 (1964), 239–341.[2] B. D. Carter and M. D. Springer, The distribution of products, quotients and powers of independent H-function variates. SIAM J. Appl. Math. 33 (1977), no. 4, 542–558.[3] J.-F. Chamayou and G. Letac, Additive properties of the Dufresne laws and their multivariate extension. J. Theoret. Probab. 12 (1999), no. 4, 1045–1066.[4] D. Dufresne, The beta product distribution with complex parameters. Comm. Statistics – Theory and Methods 39 (2010), no. 5, 837–854.[5] D. Dufresne, G distributions and the beta-gamma algebra. Preprint, University of Melbourne, 2009.[6] C. Fox, The G and H functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc. 98 (1961) 395–429.[7] M. Kaluszka and W. Krysicki, On decompositions of some random variables. Metrika 46 (1997), no. 2, 159–175.[8] A. M. Mathai and R. K. Saxena, On the linear combinations of stochastic variables. 20 (1973), 160–169.[9] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Lecture Notes in Mathematics, Vol. 348, Springer-Verlag, Berlin-New York, 1973.[10] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function. Theory and Applications. Springer, New York, 2010. xiv+268 pp. ISBN: 978-1-4419-0915-2[11] C. S. Meijer, On the G-function. I–VIII. Nederl. Akad. Wetensch., Proc. 49, (1946) 227–237, 344–356, 457–469, 632–641, 765–772, 936–943, 1063–1072, 1165–1175 = Indagationes Math. 8 (1946), 124–134, 213–225, 312–324, 391–400, 468–475, 595–602, 661–670, 713–723.[12] E. W. Weisstein, Fox H-Function. MathWorld. http://mathworld.wolfram.com/FoxH-Function.html[13] E. W. Weisstein, Meijer G-Function. MathWorld. http://mathworld.wolfram.com/MeijerG-Function.html

2010/12/30 - 03:10