The Horton laws originated in hydrology with a 1945 paper by Robert E. Horton, and for a long time remained a purely empirical finding. Ubiquitous in hierarchical branching systems, the Horton laws have been rediscovered in many disciplines ranging from geomorphology to genetics to computer science. Attempts to build a mathematical foundation behind the Horton laws during the 1990s revealed their close connection to the operation of pruning – erasing a tree from the leaves down to the root. This survey synthesizes recent results on invariances and self-similarities of tree measures under various forms of pruning. We argue that pruning is an indispensable instrument for describing branching structures and representing a variety of coalescent and annihilation dynamics. The Horton laws appear as a characteristic imprint of self-similarity, which settles some questions prompted by geophysical data.

In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no–information, rank–dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.

We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by presenting a single assumption and theorem that together unify and strengthen many tail bounds for martingales, including classical inequalities (1960–80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980–2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Peña; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. In doing so, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cramér-Chernoff method, self-normalized processes, and other parts of the literature.

Many random recursive discrete structures may be described by a single generic model. Adopting this perspective allows us to elegantly prove limits for these structures as instances of general underlying principles, and describe their phase diagrams using a unified terminology.

We illustrate this by a selection of examples. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a local weak limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest $2$-connected component in random graphs from planar-like classes. We prove local convergence of random $k$-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their $2$-connected components.

Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.

This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^{p}$- and $\phi $-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapień, McConnell, and Woyczyński. The vast majority of the assertions presented in the paper is done under the necessary and sufficient UMD assumption on the corresponding Banach space.

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on $\mathbb{R}^{d}$, which is:

Can a given random vector $\bm{X}=(X_{1},\ldots ,X_{d})$ be represented in distribution as the first $d$ members of an infinite exchangeable sequence of random variables?

This is the case if and only if $\bm{X}$ has a stochastic representation that is “conditionally iid” according to the seminal de Finetti’s Theorem. Of particular interest are cases in which the original motivation behind the model $\bm{X}$ is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when $\bm{X}$ takes values in $\{0,1\}^{d}$, has a spherical law, a law with $\ell _{1}$-norm symmetric survival function, or a law with $\ell _{\infty }$-norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on $\mathbb{R}$, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when $\bm{X}$ has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Lévy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than $\mathbb{R}$. Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.