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.

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.

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 $f : X \rightarrow X$ 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 $\Lambda^{\mathbb{Z}}$, where $\Lambda$ is a finite alphabet, and a continuous surjection $\pi : Y \rightarrow X$ 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 $\upsilon$ supported by $Y$ in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure $\mu$ on $X$, the “Lyapunov exponent” of $\mu$ (the genuine Lyapunov exponent for ergodic $\mu$, 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.

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.

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.

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.