We study the local limit in distribution of Bienaymé–Galton–Watson trees conditioned on having large sub-populations. Assuming a generic and aperiodic condition on the offspring distribution, we prove the existence of a limit given by a Kesten’s tree associated with a certain critical offspring distribution.

Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting measure of particles of generation n. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of $(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of $(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and do not rely on Martin boundary theory.

We consider the random series–parallel graph introduced by Hambly and Jordan (2004 Adv. Appl. Probab. 36, 824–838), which is a hierarchical graph with a parameter $p\in [0, \, 1]$. The graph is built recursively: at each step, every edge in the graph is either replaced with probability p by a series of two edges, or with probability $1-p$ by two parallel edges, and the replacements are independent of each other and of everything up to then. At the nth step of the recursive procedure, the distance between the extremal points on the graph is denoted by $D_n (p)$. It is known that $D_n(p)$ possesses a phase transition at $p=p_c \;:\!=\;\frac{1}{2}$; more precisely, $\frac{1}{n}\log {{\mathbb{E}}}[D_n(p)] \to \alpha(p)$ when $n \to \infty$, with $\alpha(p) >0$ for $p>p_c$ and $\alpha(p)=0$ for $p\le p_c$. We study the exponent $\alpha(p)$ in the slightly supercritical regime $p=p_c+\varepsilon$. Our main result says that as $\varepsilon\to 0^+$, $\alpha(p_c+\varepsilon)$ behaves like $\sqrt{\zeta(2) \, \varepsilon}$, where $\zeta(2) \;:\!=\; \frac{\pi^2}{6}$.

The continuous random energy model (CREM) was introduced by Bovier and Kurkova in 2004 as a toy model of disordered systems. Among other things, their work indicates that there exists a critical point $\beta_\mathrm{c}$ such that the partition function exhibits a phase transition. The present work focuses on the high-temperature regime where $\beta<\beta_\mathrm{c}$. We show that, for all $\beta<\beta_\mathrm{c}$ and for all $s>0$, the negative s moment of the CREM partition function is comparable with the expectation of the CREM partition function to the power of $-s$, up to constants that are independent of N.

We consider a stochastic model, called the replicator coalescent, describing a system of blocks of k different types that undergo pairwise mergers at rates depending on the block types: with rate $C_{ij}\geq 0$ blocks of type i and j merge, resulting in a single block of type i. The replicator coalescent can be seen as a generalisation of Kingman’s coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called replicator equations from evolutionary game theory. By dilating time arbitrarily close to zero, we see that initially, on coming down from infinity, the replicator coalescent behaves like the solution to a certain replicator equation. Thereafter, stochastic effects are felt and the process evolves more in the spirit of a multi-type death chain.

We prove an ergodic theorem for Markov chains indexed by the Ulam–Harris–Neveu tree over large subsets with arbitrary shape under two assumptions: (i) with high probability, two vertices in the large subset are far from each other, and (ii) with high probability, those two vertices have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that these assumptions are satisfied for some usual trees. Finally, with Markov chain Monte Carlo considerations in mind, we prove that when the underlying Markov chain is stationary and reversible, the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes. In doing so, we prove that the Hosoya–Wiener polynomial is minimized over $[{-}1,1]$ by the line graph among trees of a given size.

We analyse a Markovian SIR epidemic model where individuals either recover naturally or are diagnosed, leading to isolation and potential contact tracing. Our focus is on digital contact tracing via a tracing app, considering both its standalone use and its combination with manual tracing. We prove that as the population size n grows large, the epidemic process converges to a limiting process, which, unlike with typical epidemic models, is not a branching process due to dependencies created by contact tracing. However, by grouping to-be-traced individuals into macro-individuals, we derive a multi-type branching process interpretation, allowing computation of the reproduction number R. This is then converted to an individual reproduction number $R^\mathrm{(ind)}$, which, in contrast to R, decays monotonically with the fraction of app-users, while both share the same threshold at 1. Finally, we compare digital (only) contact tracing and manual (only) contact tracing, proving that the critical fraction of app-users, $\pi_{\mathrm{c}}$, required for $R=1$ is higher than the critical fraction manually contact-traced, $p_{\mathrm{c}}$, for manual tracing.

In this paper, we study asymptotic behaviors of a subcritical branching Brownian motion with drift $-\rho$, killed upon exiting $(0, \infty)$, and offspring distribution $\{p_k{:}\; k\ge 0\}$. Let $\widetilde{\zeta}^{-\rho}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-\rho}$ the maximal position of all the particles alive at time t and $\widetilde{M}^{-\rho}:\!=\max_{t\ge 0}\widetilde{M}_t^{-\rho}$ the all-time maximal position. Let $\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at $x\in (0,\infty)$. Under the assumption $\sum_{k=1}^\infty k ({\log}\; k) p_k <\infty$, we establish the decay rates of $\mathbb{P}_x(\widetilde{\zeta}^{-\rho}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-\rho}>y)$ as t and y respectively tend to $\infty$. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-\rho}> z(t,\rho))$ as $t\to\infty$, where $z(t,\rho)=\sqrt{t}z-\rho t$ for $\rho\leq 0$ and $z(t,\rho)=z$ for $\rho>0$. As a consequence, we obtain a Yaglom-type limit theorem.

We consider a superprocess $\{X_t\colon t\geq 0\}$ in a random environment described by a Gaussian field $\{W(t,x)\colon t\geq 0,x\in \mathbb{R}^d\}$. First, we set up a representation of $\mathbb{E}[\langle g, X_t\rangle\mathrm{e}^{-\langle \,f,X_t\rangle }\mid\sigma(W)\vee\sigma(X_r,0\leq r\leq s)]$ for $0\leq s < t$ and some functions f,g, which generalizes the result in Mytnik and Xiong (2007, Theorem 2.15). Next, we give a uniform upper bound for the conditional log-Laplace equation with unbounded initial values. We then use this to establish the corresponding conditional entrance law. Finally, the excursion representation of $\{X_t\colon t\geq 0\}$ is given.

This paper is concerned with the growth rate of susceptible–infectious–recovered epidemics with general infectious period distribution on random intersection graphs. This type of graph is characterised by the presence of cliques (fully connected subgraphs). We study epidemics on random intersection graphs with a mixed Poisson degree distribution and show that in the limit of large population sizes the number of infected individuals grows exponentially during the early phase of the epidemic, as is generally the case for epidemics on asymptotically unclustered networks. The Malthusian parameter is shown to satisfy a variant of the classical Euler–Lotka equation. To obtain these results we construct a coupling of the epidemic process and a continuous-time multitype branching process, where the type of an individual is (essentially) given by the length of its infectious period. Asymptotic results are then obtained via an embedded single-type Crump–Mode–Jagers branching process.

We show that for $\lambda\in[0,{m_1}/({1+\sqrt{1-{1}/{m_1}}})]$, the biased random walk’s speed on a Galton–Watson tree without leaves is strictly decreasing, where $m_1\geq 2$. Our result extends the monotonic interval of the speed on a Galton–Watson tree.

We establish a number of results concerning the limiting behaviour of the longest edges in the genealogical tree generated by a continuous-time Galton–Watson process. Separately, we consider the large-time behaviour of the longest pendant edges, the longest (strictly) interior edges, and the longest of all the edges. These results extend the special case of long pendant edges of birth–death processes established in Bocharov et al. (2023).

The study of many population growth models is complicated by only partial observation of the underlying stochastic process driving the model. For example, in an epidemic outbreak we might know when individuals show symptoms to a disease and are removed, but not when individuals are infected. Motivated by the above example and the long-established approximation of epidemic processes by branching processes, we explore the number of individuals alive in a time-inhomogeneous branching process with a general phase-type lifetime distribution given only (partial) information on the times of deaths of individuals. Deaths are detected independently with a detection probability that can vary with time and type. We show that the number of individuals alive immediately after the kth detected death can be expressed as the mixture of random variables each of which consists of the sum of k independent zero-modified geometric distributions. Furthermore, in the case of an Erlang lifetime distribution, we derive an easy-to-compute mixture of negative binomial distributions as an approximation of the number of individuals alive immediately after the kth detected death.

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in a time-inhomogeneous random environment (RWRE), which is related to the BRWRE by the many-to-one formula. The key step is to figure out Tanaka’s decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting in itself.

We introduce a modification of the generalized Pólya urn model containing two urns, and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$ added to the urns after n draws, where $J\in\mathbb{N}$. We provide sufficient conditions under which the random variables $(B_j(n))_{n\in\mathbb{N}}$, properly normalized and centered, converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second-order behavior occurs at $\sqrt{\rho}$ and not at $\rho/2$, where $\rho$ is the dominant eigenvalue of the mean replacement matrix.

We investigate the tail behavior of the first-passage time for Sinai’s random walk in a random environment. Our method relies on the connection between Sinai’s walk and branching processes with immigration in a random environment, and the analysis on some important quantities of these branching processes such as extinction time, maximum population, and total population.

We consider linear-fractional branching processes (one-type and two-type) with immigration in varying environments. For $n\ge0$, let $Z_n$ count the number of individuals of the nth generation, which excludes the immigrant who enters the system at time n. We call n a regeneration time if $Z_n=0$. For both the one-type and two-type cases, we give criteria for the finiteness or infiniteness of the number of regeneration times. We then construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. For example, it may happen that the process is extinct, but there are only finitely many regeneration times. We also study the asymptotics of the number of regeneration times of the model in the example.

We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists for which the process is stationary and ergodic, and for any other initialization the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

We investigate some aspects of the problem of the estimation of birth distributions (BDs) in multi-type Galton–Watson trees (MGWs) with unobserved types. More precisely, we consider two-type MGWs called spinal-structured trees. This kind of tree is characterized by a spine of special individuals whose BD $\nu$ is different from the other individuals in the tree (called normal, and whose BD is denoted by $\mu$). In this work, we show that even in such a very structured two-type population, our ability to distinguish the two types and estimate $\mu$ and $\nu$ is constrained by a trade-off between the growth-rate of the population and the similarity of $\mu$ and $\nu$. Indeed, if the growth-rate is too large, large deviation events are likely to be observed in the sampling of the normal individuals, preventing us from distinguishing them from special ones. Roughly speaking, our approach succeeds if $r\lt \mathfrak{D}(\mu,\nu)$, where r is the exponential growth-rate of the population and $\mathfrak{D}$ is a divergence measuring the dissimilarity between $\mu$ and $\nu$.