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.

The normalised partial sums of values of a nonnegative multiplicative function over divisors with appropriately restricted lengths of a random permutation from the symmetric group define trajectories of a stochastic process. We prove a functional limit theorem in the Skorokhod space when the permutations are drawn uniformly at random. Furthermore, we show that the paths of the limit process almost surely belong to the space of continuous functions on the unit interval and, exploiting results from number-theoretic papers, we obtain rather complex formulas for the limits of joint power moments of the process.

We study quasi-stationary distributions and quasi-limiting behaviour of Markov chains in general reducible state spaces with absorption. First, we consider state spaces that can be decomposed into two successive subsets (with communication possible in a single direction), differentiating between three situations, and characterize the exponential order of magnitude and the exact polynomial correction, called the polynomial convergence parameter, for the leading-order term of the semigroup for large time. Second, we consider general Markov chains with finitely or countably many communication classes by applying the first results iteratively over the communication classes of the chain. We conclude with an application of these results to the case of denumerable state spaces, where we prove existence for a quasi-stationary distribution without assuming irreducibility before absorption, but only aperiodicity, existence of a Lyapunov function, and existence of a point with almost surely finite return time.

In this paper we are concerned with susceptible–infected–removed (SIR) epidemics with vertex-dependent recovery and infection rates on complete graphs. We show that the hydrodynamic limit of our model is driven by a nonlinear function-valued ordinary differential equation consistent with a mean-field analysis. We further show that the fluctuation of our process is driven by a generalized Ornstein–Uhlenbeck process. A key step in the proofs of the main results is to show that states of different vertices are approximately independent as the population $N\rightarrow+\infty$.

In this article, we investigate the quantitative law of large numbers for noncommutative random variables. Firstly, we establish a Baum–Katz theorem for noncommutative successively independent random variables, which resolves an open problem posed by Stoica [48]. Our approach differs from the classical treatment but relies on the theory of asymmetric maximal inequality for noncommutative martingales. Additionally, we derive a moderate deviation inequality for noncommutative successively independent sequences, and extend this result together with the Baum–Katz theorem to noncommutative martingales. Finally, we conclude the article by applying our results to derive a noncommutative Marcinkiewicz–Zygmund-type strong laws of large numbers theorem, which extends the result of Łuczak [37] in some aspects.

The money exchange model is a type of agent-based model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model, the uniform reshuffling model, and the uniform saving model, all of which are types of money exchange model, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that, under appropriate scaling, the asymptotic wealth distribution converges to an exponential distribution for the uniform reshuffling model, and to either an exponential distribution or a gamma distribution depending on the tail behavior of the number of coins given/saved in the immediate exchange model and the random saving model, which generalizes the uniform saving model. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.

General additive functionals of patricia tries are studied asymptotically in a probabilistic model with independent, identically distributed letters from a finite alphabet. Asymptotic normality is shown after normalization together with asymptotic expansions of the moments. There are two regimes depending on the algebraic structure of the letter probabilities, with and without oscillations in the expansion of moments. As applications firstly the proportion of fringe trees of patricia tries with k keys is studied, which is oscillating around $(1-\rho(k))/(2H)k(k-1)$, where H denotes the source entropy and $\rho(k)$ is exponentially decreasing. The oscillations are identified explicitly. Secondly, the independence number of patricia tries and of tries is considered. The general results for additive functions also apply, where a leading constant is numerically approximated. The results extend work of Janson on tries by relating additive functionals on patricia tries to additive functionals on tries.

We establish large deviations for dynamical Schrödinger problems driven by perturbed Brownian motions when the noise parameter tends to zero. Our results show that Schrödinger bridges charge exponentially small masses outside the support of the limiting law that agrees with the optimal solution to the dynamical Monge–Kantorovich optimal transport problem. Our proofs build on mixture representations of Schrödinger bridges and establishing exponential continuity of Brownian bridges with respect to the initial and terminal points.

We study a queueing system with a fixed number of parallel service stations of infinite servers, each having a dedicated arrival process, and one flexible arrival stream that is routed to one of the service stations according to a ‘weighted’ shortest queue policy. We consider the model with general arrival processes and general service time distributions. Assuming that the dedicated arrival rates are of order n and the flexible arrival rate is of order $\sqrt{n}$, we show that the diffusion-scaled queueing processes converge to a stochastic Volterra integral equation with ‘ranks’ driven by a continuous Gaussian process. It reduces to the limiting diffusion with a discontinuous drift in the Markovian setting.

We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of $({a_T-1})/{T{\textrm{e}}^{b_TTx}}$, the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of $1/T^2$, the sequence converges in law to an integrated Cox–Ingersoll–Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.