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.

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.

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.

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.