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.
In this note we provide a short proof of the distributional equality between last passage percolation with geometric weights along a general down-right path and Schur processes. We do this in both the full-space and half-space settings, and for general parameters. The main inputs for our arguments are generalizations of the Robinson-Schensted-Knuth correspondence and Greene's theorem due to Krattenthaler, which are based on Fomin's growth diagrams.
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.
