The topic of Poisson approximation has attracted a lot of attention in the past seven decades. It has applications in insurance, extreme value theory, reliability theory, mathematical biology, etc.. The survey is devoted to the methods of Poisson approximation to the distribution of a sum of random variables. The methods appear useful when one deals with the problem of compound Poisson approximation as well. We formulate a number of open problems and discuss directions of further research.
Monte Carlo methods stand as a prominent computational technique extensively applied across a diverse array of disciplines in both natural and social sciences. Monte Carlo methods draw theoretical support from the strong law of large numbers and the central limit theorem, respectively, ensuring the convergence of estimates to the true mean and the weak convergence of the error to a normal distribution. While both theoretical foundations are commonly taken for granted, often without much skepticism, it is not very uncommon to encounter scenarios where the estimator possesses infinite variance, often without being readily apparent, thereby hindering the application of the central limit theorem and thus leading to unstable estimation convergence and the invalidity of confidence intervals, often in a subtle or obscured fashion. The present article aims to provide an accessible entry point into the exploration of diverse research avenues concerning Monte Carlo methods with infinite variances. It endeavors to elucidate those often overlooked yet critical instances, offering insightful discussions on feasible strategies and inherent limitations. Accompanied by a variety of relevant remarks and expositorial examples throughout, we aim to illuminate the potential, limitations and subtleties of Monte Carlo methods in applications when the estimator variance is infinite.
This survey aims to review two decades of progress on exponential functionals of (possibly killed) real-valued L & eacute;vy processes. Since the publication of the seminal survey by Bertoin and Yor, substantial advances have been made in understanding the structure and properties of these random variables. At the same time, numerous applications of these quantities have emerged across various different contexts of modern applied probability. Motivated by all this, in this manuscript, we provide a detailed overview of these developments, beginning with a discussion of the class of special functions that have played a central role in recent progress, and then organising the main results on exponential functionals into thematic groups. Moreover, we complement several of these results and set them within a unified framework. Throughout, we strive to offer a coherent historical account of each contribution, highlighting both the probabilistic and analytical techniques that have driven the advances in the field.
A nonnegative function on the vertices of an infinite graph G which vanishes at a distinguished vertex o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We survey basic properties of potentials in recurrent networks. In particular, we show that potentials are Lipschitz with respect to the effective resistance metric, and if the potential is unique, then there is a determinantal formula for the harmonic measures from infinity. We also infer from the von Neumann minimax theorem that there always exists a potential tending to infinity.
We consider a probability measure on cycle-rooted spanning forests (CRSFs) introduced by Kenyon. CRSFs are spanning subgraphs, each connected component of which has a unique cycle; they generalize spanning trees. A generalization of Wilson's celebrated CyclePopping algorithm for uniform spanning trees has been proposed for CRSFs, and several concise proofs have been given that the algorithm samples from Kenyon's distribution. In this survey, we flesh out all the details of such a proof of correctness, progressively generalizing a proof by Marchal for spanning trees. This detailed proof has several interests. First, it serves as a modern tutorial on Wilson's algorithm, suitable for applied probability and computer science audiences. Compared to uniform spanning trees, the more sophisticated motivating application to CRSFs brings forth connections to recent research topics such as loop measures, partial rejection sampling, and heaps of cycles. Second, the detailed proof & agrave; la Marchal yields the law of the time complexity of the sampling algorithm, shedding light on practical situations where the algorithm is expected to run fast.
In this survey paper, we revisit the sample-path moderate deviation principles (MDPs) for random walks on the real line. The aim is to provide an overview of the existing techniques that can be used to prove these results. We focus on two approaches: (i) MDP for finite dimensional distributions and exponential tightness in a function space with an appropriate topology, and (ii) variational method via Donsker-Varadhan variational formula. In particular, under weak tail conditions on the distribution of the i.i.d. variables, using both approaches, we prove the sample-path MDP in C under uniform topology and in D under the Skorohod J1 topology.
We give an overview of various results and methods related to information-theoretic distances of R & eacute;nyi type in the light of their applications to the central limit theorem (CLT). The first part (Sections 1-9) is devoted to the total variation and the Kullback-Leibler distance (relative entropy). In the second part (Sections 10-15) we discuss general properties of R & eacute;nyi and Tsallis divergences of order alpha > 1, and then in the third part (Sections 16-21) we turn to the CLT and non-uniform local limit theorems with respect to these strong distances. In the fourth part (Sections 22-31), we discuss recent results on strictly subgaussian distributions and describe necessary and sufficient conditions which ensure the validity of the CLT with respect to the R & eacute;nyi divergence of infinite order.
We consider directed random graphs, the prototype of which being the Barak-Erdos graph (G) over right arrow (Z, p), and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the A number of limit theorems are shown. The goal of this paper is to present results along with techniques that have been used in this area. In the case of (G) over right arrow (Z, p) the last passage percolation constant C(p) is studied in great detail. It is shown that C(p) is analytic for p > 0, has an interesting asymptotic expansion at p = 1 and that C(p)/p converges to e like 1/(log p)(2) as p -> 0. The paper includes the study of IBMs as models on their own as well as their connections to stochastic models of branching processes in continuous or discrete time with selection. Several proofs herein are new or simplified versions of published ones. Regenerative techniques are used where possible, exhibiting random sets of vertices over which the graphs regenerate. When edges have random weights we show how the last passage percolation constants behave and when central limit theorems exist. When the underlying vertex set is partially ordered, new phenomena occur, e.g., there are relations with last passage Brownian percolation. We also look at weights that may possibly take negative values and study in detail some special cases that require combinatorial/graph theoretic techniques that exhibit some interesting non-differentiability properties of the last passage percolation constant. In addition, we explain how to approach the problem of estimation of last passage percolation constants by means of perfect simulation.
Activated Random Walk is a particle system displaying Self-Organized Criticality, in that the dynamics spontaneously drive the system to a critical state. How universal is this critical state? We state many interlocking conjectures aimed at different aspects of this question: scaling limits, microscopic limits, temporal and spatial mixing, incompressibility, and hyperuniformity.
In this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [28]) and L. Gross (see [56]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus.
This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such as Eldan's stochastic localisation and the F\"ollmer process, the Bou\'e--Dupuis variational formula and the Barashkov--Gubinelli approach, the transportation of measure perspective, and the classical analogues of these ideas for Hamilton--Jacobi equations which arise in mean-field limits.
Partial Rejection Sampling is an algorithmic approach to obtaining a perfect sample from a specified distribution. The objects to be sampled are assumed to be represented by a number of random variables. In contrast to classical rejection sampling, in which all variables are resampled until a feasible solution is found, partial rejection sampling aims at greater efficiency by resampling only a subset of variables that `go wrong'. Partial rejection sampling is closely related to Moser and Tardos' algorithmic version of the Lov\'asz Local Lemma, but with the additional requirement that a specified output distribution should be met. This article provides a largely self-contained account of the basic form of the algorithm and its analysis.
A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, L\'evy processes, L\'evy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a L\'evy process -, strong Markov processes and Gaussian processes. Using the Burkholder-Davis-Gundy inequalities we als discuss some relations between maximal estimates in probability and the Hardy-Littlewood maximal functions from analysis. This paper has been accepted for publication in Probability Surveys
In this article, I present different definitions of elementary pro-cesses which lead to definitions of Ito integrals against cylindrical Wiener processes. I prove the equivalence between those definitions by construct-ing this Ito integral using only Ito isometries and extensions by density. Then, from the perspective of white noise theory and Kondratiev spaces, I compare the definitions of cylindrical Wiener processes, and present a link between Ito integral and cylindrical white noise.
In the last two decades there was a lot of progress in understanding the geometry of smooth Gaussian fields. This survey aims to cover one particular line of research: the large scale behaviour of level and excursion sets and their (conjectured) connection to Bernoulli percolation.
Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of $G$ with $n$ nodes, for some $n\leq N$ (the spanning tree case correspond to $n=N$, and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs $G$, and any $n\leq N$. We highlight the case of the uniform subtree of $\mathbb{Z}^2$ with $n$ nodes, containing the origin $(0,0)$ for which Schramm asked several questions. We produce pictures, statistics, and some conjectures. A second aim of the paper is devoted to surveying other models of random subtrees of a graph, among them, DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide new models, some statistics, and some conjectures.
In this survey article, we present an overview of a large class of probabilistic representations of the fragmentation equation, and we develop and study the interconnections in between these representations. We focus on the stochastic process which represents the evolution of the mass of a typical particle which undergoes fragmentation in time. These probabilistic representations range from Markov chains to stochastic differential equa-tions with jumps, and we aim at constructing how they are inter-related. In particular, we show how these representations can be used to develop easy to implement numerical methods.
This article is a survey of the results on asymptotic behav-ior of small ball probabilities in L2-norm. Recent progress in this field is mainly based on the methods of spectral theory of differential and integral operators.
In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the `natural' class of processes for many concepts first developed in the Markovian framework. As an example, stochastic differential equations have been invented as a tool to study Markov processes but nowadays are treated separately in the literature. Moreover, the killing of processes has been known for decades before it made its way to the theory of semimartingales most recently. We describe, when these and other important concepts have been invented in the theory of Markov processes and how they were transferred to semimartingales. Further topics include the symbol, characteristics and generalizations of Blumenthal-Getoor indices. Some additional comments on relations between Markov processes and semimartingales round out the paper.
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often large N universal forms for this correlation after smoothing, which results in particularly simple limiting formulas for the fluctuation of the linear statistics. We review these limiting formulas, derived in the simplest cases as corollaries of explicit knowledge of the truncated two-point correlation. One of the large N limits is to scale the eigenvalues so that limiting support is compact, and the linear statistics vary on the scale of the support. This is a global scaling. The other, where a thermodynamic limit is first taken so that the spacing between eigenvalues is of order unity, and then a scale imposed on the test functions so they are slowly varying, is the bulk scaling. The latter was already identified as a probe of random matrix characteristics for quantum spectra in the pioneering work of Dyson and Mehta.
