We consider a Lévy process reflected at the origin with additional independent and identically distributed collapses that occur at Poisson epochs, where a collapse is a jump downward to a state which is a random fraction of the state just before the jump. We first study the general case, then specialize to the case where the Lévy process is spectrally positive, and, finally, we specialize further to the two cases where the Lévy process is a Brownian motion and a compound Poisson process with exponential jumps minus a linear slope.

In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.

The purpose of this paper is to analyze the degree index and the clustering index in dense random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this paper. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős–Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. In addition to the Erdős–Rényi case, we also present a simulation-based analysis for random regular graphs, the Barabási–Albert model, and the Watts–Strogatz model.

We consider general discrete-time multitype branching processes on a countable set X. According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in X, not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalising the germ order introduced by Hutchcroft, which relies on the generating function of the process. We prove that given two multitype branching processes with laws ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ respectively, with ${\boldsymbol{\mu}}\ge{\boldsymbol{\nu}}$, then in every set where there is survival according to ${\boldsymbol{\nu}}$, there is also survival according to ${\boldsymbol{\mu}}$. Moreover, in every set where there is strong survival according to ${\boldsymbol{\nu}}$, there is also strong survival according to ${\boldsymbol{\mu}}$, provided that the supremum of the global extinction probabilities for the ${\boldsymbol{\nu}}$ process, taken over all starting points x, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.

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.

We consider the asymptotic behaviour of the expectation of the perimeter deviation of a uniform random spherical disc–polygon in a spherical spindle convex disc with smooth boundary. We also introduce the notion of duality on the sphere, define a model of random circumscribed disc–polygons, and determine some asymptotic results about them.

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.

Consider a random walk in a time-inhomogeneous random environment. When the environment is stationary and ergodic, we identify a quenched harmonic function for almost every realization of the environment. This function allows us to define a random walk in a random environment conditioned to stay positive, using Doob’s h-transform.

We consider d-dimensional stochastic differential equations (SDEs) of the form $\textrm{d}U_t = b(U_t)\,\textrm{d}t + \sigma\,\textrm{d}Z_t$. Let $X_t$ denote the solution if the driving noise $Z_t$ is a d-dimensional rotationally symmetric $\alpha$-stable process ($1\lt \alpha\lt 2$), and let $Y_t$ be the solution if the driving noise is a d-dimensional Brownian motion. Continuing the work started in Deng et al. (2025), we derive an estimate of the total variation distance $\|\textrm{law}(X_{t})-\textrm{law}(Y_{t})\|_\textrm{TV}$ for all $t \gt 0$, and we show that the ergodic measures $\mu_\alpha$ and $\mu_2$ of $X_t$ and $Y_t$, respectively, satisfy $\|\mu_\alpha-\mu_2\|_\textrm{TV} \leq {Cd\log(1+d)}(2-\alpha)/({\alpha-1})$. We show that this bound is optimal with respect to $\alpha$ by an Ornstein–Uhlenbeck SDE. Combining this bound with a recent interpolation result from Huang et al. (2023), we can derive a bound in the Wasserstein-p distance ($0 \lt p \lt 1$): $\|\mu_\alpha-\mu_2\|_{W_p} \leq {Cd^{(p+3)/2}\log(1+d)}(2-\alpha)/{\alpha-1}$.

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 shock models governed by the bivariate geometric counting process. By assuming the competing risks framework, failures are due to one of two mutually exclusive causes (shocks). We obtain and study some relevant functions, such as failure densities, survival functions, probability of the cause of failure, and moments of the failure time conditioned on a specific cause. Such functions are specified by assuming that systems or living organisms fail at the first instant in which a random threshold is reached by the sum of received shocks. Under this failure scheme, various cases arising for suitable choices of the random threshold are provided too.

In recent works on the theory of machine learning, it has been observed that heavy tail properties of stochastic gradient descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, Gürbüzbalaban et al. (2021) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_n=A_nX_{n-1}+B_n$ for independent and identically distributed pairs $(A_n,B_n)$, where $A_n$ is a random symmetric matrix and $B_n$ is a random vector. However, their approach is not completely correct and, in the present paper, the problem is put into the right framework by applying the theory of irreducible-proximal matrices.

We investigate the EM approximation for $\mathbb{R}^d$-valued ergodic stochastic differential equations (SDEs) driven by rotationally invariant $\alpha$-stable processes ($\alpha\in(1,2)$) with Markovian switching. The coefficient g violates the dissipative condition for certain states of the switching process. Using the Lindeberg principle, we establish quantitative error bounds between the original process $(X_t,R_t)_{t\geqslant 0}$ and its Euler–Maruyama (EM) scheme under a specially designed metric. Furthermore, we derive both a central limit theorem and a moderate derivation principle for the empirical measures of both the SDE and its EM scheme. The theoretical results are subsequently validated through a concrete example.

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 consider the filtering problem associated with partially observed McKean–Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times; the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem is challenging even in the ordinary SDE case and requires numerical approximations. Based on the ideas in Ben Rached et al. (2024) and dos Reis et al. (2023), we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the aforementioned expectations. We prove under assumptions that, for $\varepsilon>0$, to obtain a mean square error of $\mathcal{O}(\varepsilon^2)$ the PF has a cost per observation time of $\mathcal{O}(\varepsilon^{-5})$, and the MLPF costs $\mathcal{O}(\varepsilon^{-4})$ (best case) or $\mathcal{O}(\varepsilon^{-4}\log(\varepsilon)^2)$ (worst case). Our theoretical results are supported by numerical experiments.

A random variable $\xi$ has a light-tailed distribution (for short, is light-tailed) if it possesses a finite exponential moment, ${\mathbb{E}} \, {\exp}{(\lambda \xi)} <\infty$ for some $\lambda >0$, and has a heavy-tailed distribution (is heavy-tailed) if ${\mathbb{E}} \, {\exp}{(\lambda\xi)} = \infty$ for all $\lambda>0$. Leipus et al. (2023 AIMS Math. 8, 13066–13072) presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. We show that this phenomenon is universal: any light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables. Moreover, a more general fact holds: these two independent random variables may have as heavy-tailed distributions as we wish. Further, we extend the latter result to the minimum of any finite number of independent random variables. We also comment on possible generalizations of our result to the case of dependent random variables.

We consider two-person zero-sum semi-Markov games with incomplete reward information on one side under the expected discount criterion. First, we prove that the value function exists and satisfies the Shapley equation. From the Shapley equation, we construct an optimal policy for the informed player. Second, to show the existence of an optimal policy for the uninformed player, we introduce an auxiliary dual game and establish the relationship between the primal game and the dual game. By this relationship, we also prove the existence of the value function of the dual game, and then construct an optimal policy for the uninformed player in the primal game. Finally, we develop two iterative algorithms to compute $\varepsilon$-optimal policies for the informed player and the uninformed player, respectively.

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.

We study bond and site Bernoulli percolation models on $\mathbb{Z}^d$ for $d \geq 3$ with parameter p, in both the oriented and non-oriented versions. The main macroscopic quantity of interest is the probability of long-range order, and the existence of a non-trivial threshold is well established. Precise numerical results for the threshold values are available in the literature, but mathematically rigorous bounds are mostly restricted to two-dimensional lattices. Utilizing dynamical coupling techniques, we introduce a comprehensive set of new rigorous upper bounds that corroborate existing numerical values.