Log-concavity of a joint survival function is proposed as a model for bivariate increasing failure rate (BIFR) distributions. Its connections with or distinctness from other notions of BIFR are discussed. A necessary and sufficient condition for a bivariate survival function to be log-concave (BIFR-LCC) is given that elucidates the impact of dependence between lifetimes on ageing. Illustrative examples are provided to explain BIFR-LCC for both positive and negative dependence.

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

In this paper we give a new flavour to what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population of constant size N, assume that each individual at time 0 carries a distinct type, or allele. Consider the joint dynamics of these N alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time t. Call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the N-Kingman coalescent. Stimulated by this result, we investigate whether it extends (i) to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times, and (ii) to the general case of Λ-Fleming‒Viot processes.

We study random relational structures that are relatively exchangeable—that is, whose distributions are invariant under the automorphisms of a reference structure ${M}$. When ${M}$ is ultrahomogeneous and has trivial definable closure, all random structures relatively exchangeable with respect to $m$ satisfy a general Aldous–Hoover-type representation. If ${M}$ also satisfies the n-disjoint amalgamation property (n-DAP) for all $n \ge 1$, then relatively exchangeable structures have a more precise description whereby each component depends locally on ${M}$.

Let $L$ be a countable language. We say that a countable infinite $L$ -structure ${\mathcal{M}}$ admits an invariant measure when there is a probability measure on the space of $L$ -structures with the same underlying set as ${\mathcal{M}}$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of ${\mathcal{M}}$ . We show that ${\mathcal{M}}$ admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in $\text{Aut}({\mathcal{M}})$ of an arbitrary finite tuple of ${\mathcal{M}}$ fixes no additional points. When ${\mathcal{M}}$ is a Fraïssé limit in a relational language, this amounts to requiring that the age of ${\mathcal{M}}$ have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

Full likelihood inference under Kingman's coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) methods have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general Λ- and Ξ-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites Λ- and Ξ-coalescents, and use them to obtain ‘approximately optimal’ IS and PAC algorithms for Λ-coalescents, yielding substantial gains in efficiency over existing methods.

Λ-coalescents model the evolution of a coalescing system in which any number of components randomly sampled from the whole may merge into larger blocks. This survey focuses on related combinatorial constructions and the large-sample behaviour of the functionals which characterize in some way the speed of coalescence.

Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radius r of convergence. It is shown that the asymptotic behaviour of symmetric compound Poisson models is mainly determined by the characteristic value θrφ′(r−). If θrφ′(r−)≥1, then the model is in the domain of attraction of the Kingman coalescent. If θrφ′(r−) < 1, then under mild regularity conditions a condensation phenomenon occurs which forces the model to be in the domain of attraction of a discrete-time Dirac coalescent. The proofs are partly based on the analytic saddle point method. They draw heavily from local limit theorems and from results of S. Janson on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analysed.

In this article we show the asymptotics of distribution and moments of the size Xn of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman n-coalescent for n → ∞. The Bolthausen-Sznitman n-coalescent is a Markov process taking states in the set of partitions of {1, …, n}, where 1, …, n are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in. We also provide exact formulae for the distribution of Xn. The main tool used is the connection of the Bolthausen-Sznitman n-coalescent with random recursive trees introduced by Goldschmidt and Martin (2005). With it, we show that Xn - 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with n - 1 customers.

In ruin theory, the conjecture given in De Vylder and Goovaerts (2000) is an open problem about the comparison of the finite time ruin probability in a homogeneous risk model and the corresponding ruin probability evaluated in the associated model with equalized claim amounts. In this paper we consider a weaker version of the conjecture and show that the integrals of the ruin probabilities with respect to the initial risk reserve are uniformly comparable.

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.

We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes and Rosenthal [12]. In order to do this,

• • We study and compare three notions of convergence of types in a stable theory: logic convergence, i.e., formula by formula, metric convergence (both already well studied) and convergence of canonical bases. In particular, we characterise א0-categorical stable theories in which the last two agree.

• • We characterise sequences that admit almost indiscernible sub-sequences.

• • We apply these tools to the theory of atomless random variables (ARV). We characterise types and notions of convergence of types as conditional distributions and weak/strong convergence thereof, and obtain, among other things, the Main Theorem of Berkes and Rosenthal.

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let ST,i be the stopped sum denoting the number of appearances of outcome 'i' in X1, …, XT, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, ST,0, ST,1, …, ST,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

We study a family of Markov processes on P (k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R + x ∏i=1 k P (k) with intensity dt ⊗ ϱν (k), where ϱν is the distribution of the paintbox based on the probability measure ν on P m, the set of ranked-mass partitions of 1, and ϱν (k) is the product measure on ∏i=1 k P (k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

Coalescents with multiple collisions (also called Λ-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Consider an infinite population with immigration labelled at each generation by N := {1, 2, …}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focusing on simple distinguished coalescents, i.e. such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0, 1] denoted by M = (Λ0, Λ1). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the Λ-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures Λ0 and Λ1 respectively specify the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time all individuals are immigrant children.

Consider a monoecious diploid population with nonoverlapping generations, whose size varies with time according to an irreducible, aperiodic Markov chain with states x1N,…,xKN, where K ≪ N. It is assumed that all matings except for selfing are possible and equally probable. At time 0 a random sample of n ≪ N genes is taken. Given two successive population sizes xjN and xiN, the numbers of gametes that individual parents contribute to offspring can be shown to be exchangeable random variables distributed as Gij. Under minimal conditions on the first three moments of Gij for all i and j, a suitable effective population size Ne is derived. Then if time is recorded in a backward direction in units of 2Ne generations, it can be shown that coalescent theory holds.

Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an F-S string of length (at least) k1 + k2, that is, (at least) k1 failures followed by (at least) k2 successes in n such trials, are studied. The associated waiting time for the rth occurrence of an F-S string of length (at least) k1 + k2 in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of F-S strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.

Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.

We study the number of collisions, Xn, of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite characteristic measure η(dx) = x−2Λ(dx). Via a coupling technique, we derive limiting laws of Xn, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of Xn include normal, stable with index 1 ≤ α < 2, and Mittag-Leffler distributions. The results apply, in particular, to the case when η is a beta(a − 2, b) distribution with parameters a > 2 and b > 0. The approach taken allows us to derive asymptotics of three other functionals of the coalescent: the absorption time, the length of an external branch chosen at random from the n external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.