Let $\{Y_{n}$, $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper we study lower deviation probabilities for $Y_{n}$. More precisely, assuming that $k,n \to \infty$ so that $k={\mathrm{o}} (n)$, we investigate the asymptotics of $\mathbb P(Y_{n} \leq k )$ and $\mathbb P(Y_{n} = k )$. Our results clarify the role of the moment conditions in the local limit theorem for $Y_n$ proved by Mellein (1982).

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of ‘one large jump’, be it for finite sums, random sums, or Lévy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive orthant $[0,\infty)^d$, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities rendered asymptotically negligible under classical regular variation. Second, we discover that depending on the structure of a particular tail event, the tail behavior of the aggregates may be characterized by more than a single large jump. Finally, we illustrate a similar phenomenon for regularly varying multivariate Lévy processes, establishing as well a relationship between regular variation of a multivariate Lévy process and multivariate regular variation of its Lévy measure on different subcones. The applicability of these results in financial and insurance risk management is discussed.

We study the asymptotic behavior of the number of crossings by a one-dimensional diffusion of a threshold where the process exhibits stickiness. We distinguish three types of crossings and show that to each type corresponds a distinct asymptotic regime for the respective number-of-crossings statistic. We introduce notions of bouncing as the symmetric counterparts to crossings and show that the corresponding number-of-bouncings statistics share the same asymptotic properties as their crossings counterparts. We first prove the results for sticky Brownian motion, then extend them to sticky–reflected Brownian motion (where only bouncing is possible) and to sticky diffusions. As an application, we propose consistent estimators for the stickiness parameter of sticky diffusions and sticky–reflected Brownian motion.

We investigate branching processes in a nearly degenerate varying environment, where the offspring distribution converges to the degenerate distribution at 1. Such processes die out almost surely; therefore, we either condition on non-extinction or add inhomogeneous immigration. Extending our one-dimensional limit results from Kevei and Kubatovics (2024), we derive functional limit theorems. In the former case, the limit process is a time-changed simple birth-and-death process on $(-\infty, \infty)$ conditioned on survival at 0, while in the latter, it is a time-changed stationary continuous-time Markov branching process with immigration.

We study a nonlinear branching diffusion process in the sense of McKean, i.e. where particles are subjected to a mean-field interaction. We consider first a strong formulation of the problem and we provide an existence and uniqueness result by using contraction arguments. Then we consider the notion of weak solution and its equivalent martingale problem formulation. In this setting, we provide a general weak existence result, as well as a propagation of chaos property, i.e. the McKean–Vlasov branching diffusion is the limit of a large-population branching diffusion process with mean-field interaction.

We study the local asymptotic behaviour of divergence-like functionals of a family of d-dimensional infinitely divisible random fields. Specifically, we derive limit theorems of surface integrals over Lipschitz manifolds for this class of fields when the region of integration shrinks to a single point. We show that in most cases, convergence stably in distribution holds after a proper normalisation. Furthermore, the limit random fields can be described in terms of stochastic integrals with respect to a Lévy basis. We additionally discuss the relationship between our results and the advective kinetic energy flux in a possibly turbulent flow.

Recent investigations have argued that there is a simple explicit representation for the Kolmogorov constant c associated with the subcritical Galton–Watson branching process. We exhibit examples showing that although this representation can be valid, it more often is not. Our work is presented in terms of the limiting conditional mean population size $\mu=c^{-1}$. The analogous quantity for the Markov branching process is denoted by $\widehat\mu$. We show that the simple representation put forward for $\widehat\mu$ in fact is an upper bound that is attained only if the offspring-number probability-generating function is quadratic. The conditional mean $\mu$ is the limit of a computable increasing sequence $(\mu_n$). Estimates of n are determined ensuring that, for any small positive number $\varepsilon$, $0\lt\mu-\mu_n\le \varepsilon$.

For an arbitrary negative Schwarzian unimodal map with a non-flat critical point, we establish the level-2 large deviation principle for empirical distributions. We also give an example of a bimodal map for which the level-2 large deviation principle does not hold.

Consider n points independently sampled from a density p of class $\mathcal{C}^2$ on a smooth compact d-dimensional submanifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the random walk visiting these points according to a transition kernel K. We study the almost sure uniform convergence of the generator of this process to the diffusive Laplace–Beltrami operator when n tends to infinity, from which we establish the convergence of the random walk to a diffusion process on the manifold. In contrast to known results, our result does not require the kernel K to be continuous, which covers the cases of walks exploring k-nearest neighbor (kNN) and geometric graphs, and convergence rates are given. The distance between the random walk generator and the limiting operator is separated into several terms: a statistical term, related to the law of large numbers, is treated with concentration tools and an approximation term that we control with tools from differential geometry. The case of kNN Laplacians is detailed. The convergence of the stochastic processes having these operators as generators is also studied, by establishing additional tightness results of their distributions on the space of càdlàg functions.

We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30(6) (2023), 1657–1679], we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension $\delta \in (0,\frac 23)$. We proved that, with overwhelming probability, the FUP with an exponent $\ge \frac 12-\frac 34\delta -\varepsilon $ holds for these discrete Cantor sets with random alphabets. In this paper, we construct random Cantor sets with dimension $\delta \in (0,\frac 23)$ in $\mathbb {R}$ via a different random procedure from the previous one used in Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30(6) (2023), 1657–1679]. We prove that, with overwhelming probability, the FUP with an exponent $\ge \frac 12-\frac 34\delta -\varepsilon $ holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.

We study the number of triangles $T_n$ in the sparse $\beta$-model on n vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of $T_n$. Next, by applying the Malliavin–Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between the normalized $T_n$ and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for $T_n$ as $n\to\infty$.

We study the asymptotic properties, in the weak sense, of regenerative processes and Markov renewal processes. For the latter, we derive both renewal-type results, also concerning the related counting process, and ergodic-type results, including the so-called $\varphi$-mixing property. This theoretical framework permits us to study the weak limit of the integral of a semi-Markov process, which can be interpreted as the position of a particle moving with finite velocities, taken for a random time according to the Markov renewal process underlying the semi-Markov one. Under mild conditions, we obtain the weak convergence to scaled Brownian motion. As a particular case, this result establishes the weak convergence of the classical generalized telegraph process.

We study large deviations for Cox–Ingersoll–Ross processes with small noise and state-dependent fast switching via associated Hamilton–Jacobi–Bellman equations. As time scales separate, when the noise goes to 0 and the rate of switching goes to $\infty$, we get a limit equation characterized by the averaging principle. Moreover, we prove the large deviation principle with an action-integral form rate function to describe the asymptotic behavior of such systems. The new ingredient is establishing the comparison principle in the singular context. The proof is carried out using the nonlinear semigroup method from Feng and Kurtz’s book [14].

Given a sequence of graphs $G_n$ and a fixed graph H, denote by $T(H, G_n)$ the number of monochromatic copies of the graph H in a uniformly random c-coloring of the vertices of $G_n$. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs $H_1, H_2, \ldots, H_d$ we derive the joint distribution of $(T(H_1, G_n^{(1)}), T(H_2, G_n^{(2)}), \ldots, T(H_d, G_n^{(d)}))$, where $\mathbf{G}_n = (G_n^{(1)}, G_n^{(2)}, \ldots, G_n^{(d)})$ is a collection of dense graphs on the same vertex set converging in the multiplex cut-metric. The limiting distribution is the sum of two independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.

We consider a critical bisexual branching process in a random environment generated by independent and identically distributed random variables. Assuming that the process starts with a large number of pairs N, we prove that its extinction time is of order $\ln^2 N$. Interestingly, this result is valid for a general class of mating functions. Among these are the functions describing the monogamous and polygamous behavior of couples, as well as the function reducing the bisexual branching process to the simple one.

In this paper, we study the self-normalized Cramér-type moderate deviation of the empirical measure of the stochastic gradient Langevin dynamics (SGLD). Consequently, we also derive the Berry–Esseen bound for the SGLD. Our approach is by constructing a stochastic differential equation to approximate the SGLD and then applying Stein’s method to decompose the empirical measure into a martingale difference series sum and a negligible remainder term.

We investigate the limiting spectral distribution of a noncentral unified matrix model defined by $\boldsymbol{\Omega}(\mathbf{X}) = ({(\mathbf{X}\mathbf{P}_1+\mathbf{A})(\mathbf{X}\mathbf{P}_1+\mathbf{A})'}/{n_1}) ({\mathbf{X}\mathbf{P}_2\mathbf{X}'}/{n_2})^{-1}$, where $\mathbf{X}=(X_{ij})_{p\times n}$ is a random matrix with independent and identically distributed real entries having zero mean and finite second moment. $\mathbf{A}$ is a $p\times n$ nonrandom matrix. The matrices $\mathbf{P}_1$ and $\mathbf{P}_2$ are projection matrices satisfying $\mathrm{rank}(\mathbf{P}_1)=n_1$, $\mathrm{rank}(\mathbf{P}_2)=n_2$, and $\mathbf{P}_1\mathbf{P}_2=0$. When $\mathbf{P}_1$ and $\mathbf{P}_2$ are random, they are assumed to be independent of $\mathbf{X}$. When $p/n_1\to c_1\in(0,\infty)$ and $p/n_2\to c_2\in(0,1)$, we establish the almost sure convergence of the empirical spectral distribution of $\boldsymbol{\Omega}$ to a deterministic limiting distribution. Furthermore, we show that this limiting distribution coincides with that of the noncentral F-matrix, thus revealing a deep connection between the proposed model and classical multivariate analysis.

We consider a generalization of the forest fire model on $\mathbb{Z}_+$ with ignition at zero only, studied by Volkov (2009 ALEA 6, 399–414). Unlike that model, we allow delays in the spread of the fires and the non-zero burning time of individual ‘trees’. We obtain some general properties for this model, which cover, among others, the phenomenon of an ‘infinite fire’, not present in the original model.

We investigate some investment problems related to maximizing the expected utility of the terminal wealth in a continuous-time Itô–Markov additive market. In this market, the prices of financial assets are described by Markov additive processes that combine Lévy processes with regime-switching models. We give explicit expressions for the solutions to the portfolio selection problem for the hyperbolic absolute risk aversion (HARA) utility, the exponential utility, and the extended logarithmic utility. In addition, we demonstrate that the solutions for the HARA utility are stable in terms of weak convergence when the parameters vary in a suitable way.

Following the pivotal work of Sevastyanov (1957), who considered branching processes with homogeneous Poisson immigration, much has been done to understand the behaviour of such processes under different types of branching and immigration mechanisms. Recently, the case where the times of immigration are generated by a non-homogeneous Poisson process has been considered in depth. In this work, we demonstrate how we can use the framework of point processes in order to go beyond the Poisson process. As an illustration, we show how to transfer techniques from the case of Poisson immigration to the case where it is spanned by a determinantal point process.