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 consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as a hypothesis-testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on n vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k=o(n)$ vertices are given spatial locations and connect following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test based on counting normalized triangles to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n\to\infty$, and identifies large-degree vertices of the community with high probability.

Any margin of the multinomial distribution is multinomially distributed. Retaining this closure property, a family of generalized multinomial distributions is proposed. This family is characterized within multiplicative probability measures, using the Bell polynomial. The retained closure property simplifies marginal properties such as moments. The family can be obtained by conditioning independent infinitely divisible distributions on the total and also by mixing the multinomial distribution with the normalized infinitely divisible distribution. The closure property justifies a stochastic process of the family by Kolmogorov’s extension theorem. Over time, Gibbs partitions of a positive integer appear as the limiting distributions of the family.

Negative dependence in tournaments has received attention in the literature. The property of negative orthant dependence (NOD) was proved for different tournament models with a special proof for each model. For general round-robin tournaments and knockout tournaments with random draws, Malinovsky and Rinott (2023) unified and simplified many existing results in the literature by proving a stronger property, negative association (NA). For a knockout tournament with a non-random draw, they presented an example to illustrate that ${\boldsymbol{S}}$ is NOD but not NA. However, their proof is not correct. In this paper, we establish the properties of negative regression dependence (NRD), negative left-tail dependence (NLTD), and negative right-tail dependence (NRTD) for a knockout tournament with a random draw and with players being of equal strength. For a knockout tournament with a non-random draw and with equal strength, we prove that ${\boldsymbol{S}}$ is NA and NRTD, while ${\boldsymbol{S}}$ is, in general, not NRD or NLTD.

We study the bilateral preference graphs $\mathit{LK}(n, k)$ of La and Kabkab, obtained as follows. Put independent and uniform [0, 1] weights on the edges of the complete graph $K_n$. Then, each edge (i, j) is included in $\mathit{LK}(n,k)$ if it is bilaterally preferred, in the sense that it is among the k edges of lowest weight incident to vertex i, and among the k edges of lowest weight incident to vertex j. We show that $k = \log(n)$ is the connectivity threshold, solving a conjecture of La and Kabkab, and obtaining finer results about the window. We also investigate the asymptotic behavior of the average degree of vertices in $\mathit{LK}(n, k)$ as $n\rightarrow\infty$.

We study backward stochastic difference equations (BS$\Delta$Es) driven by a d-dimensional stochastic process on a lattice, whose increments take only $d+1$ possible values that generate the lattice. Interpreting the driving process as a d-dimensional asset price process, we provide applications to an optimal investment problem and to a market equilibrium analysis, where utility functionals are defined via BS$\Delta$Es.

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.

Hybrid stochastic differential equations (SDEs) are a useful tool for modeling continuously varying stochastic systems modulated by a random environment, which may depend on the system state itself. In this paper we establish the pathwise convergence of solutions to hybrid SDEs using space-grid discretizations. Though time-grid discretizations are a classical approach for simulation purposes, our space-grid discretization provides a link with multi-regime Markov-modulated Brownian motions. This connection allows us to explore aspects that have been largely unexplored in the hybrid SDE literature. Specifically, we exploit our convergence result to obtain efficient and computationally tractable approximations for first-passage probabilities and expected occupation times of the solutions to hybrid SDEs. Lastly, we illustrate the effectiveness of the resulting approximations through numerical examples.

We study two continuous-time, time-inconsistent problems for an individual who purchases life annuities and invests her wealth in a risky asset under the mean-variance criterion. In the first problem, the buyer may only purchase life annuities at a bounded, continuous rate, while in the second problem, the buyer may purchase any amount of life annuity income at any time, which results in a singular control problem. We find the individual’s time-consistent equilibrium control strategies explicitly for the two life-annuity problems by solving the corresponding extended Hamilton–Jacobi–Bellman systems of equations. We also discuss the effects of parameters on the equilibrium strategies of the two life-annuity problems.

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.

This paper considers option valuation under finite mixture models in a discrete-time economy. Specifically, the Esscher transform is employed to select a pricing kernel. Novel finite mixture models with negative-shifted Gamma and negative-shifted inverse Gaussian distributions are developed. A hybrid finite mixture model that allows different parametric forms for component distributions is introduced to incorporate model uncertainty. An empirical characteristic function estimation method is employed to estimate the finite mixture models. Closed-form pricing formulas for a European call option are obtained for some finite mixture models. Empirical examples using data on the Bitcoin-USD prices are provided to illustrate an application of the proposed models to value Bitcoin options.

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.

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 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.

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.

For Markov chains and Markov processes exhibiting a form of stochastic monotonicity (higher states have higher transition probabilities in terms of stochastic dominance), stability and ergodicity results can be obtained with the use of order-theoretic mixing conditions. We complement these results by providing quantitative bounds on deviations between distributions. We also show that well-known total variation bounds can be recovered as a special case.

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.

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.