We describe the asymptotic behaviour of large degrees in random hyperbolic graphs for all values of the curvature parameter $\alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the sparse regime $\alpha>\tfrac{1}{2}$, the rank at which these two rankings cease to coincide is $n^{1/(1+8\alpha)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $\alpha = \tfrac{1}{2}$, which corresponds to the connectivity threshold of the model. For $\alpha < \tfrac{1}{2}$, the maximum degree is of order $n - O(n^{\alpha + 1/2})$, whereas for $\alpha \geq \tfrac{1}{2}$, the maximum degree is of order $n^{1/(2\alpha)}$. In the $\alpha < \tfrac{1}{2}$ and $\alpha > \tfrac{1}{2}$ cases, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $\alpha < \tfrac{1}{2}$ and Fréchet for $\alpha > \tfrac{1}{2}$). This refines previous estimates on the maximum degree for $\alpha > \tfrac{1}{2}$ and extends the study of large degrees to the dense regime $\alpha \leq \tfrac{1}{2}$.

We derive the exact asymptotics of $\mathbb{P} {\{\sup\nolimits_{\boldsymbol{t}\in {\mathcal{A}}}X(\boldsymbol{t})>u \}} \textrm{ as}\ u\to\infty,$ for a centered Gaussian field $X({\boldsymbol{t}}),\ {\boldsymbol{t}}\in \mathcal{A}\subset\mathbb{R}^n$, $n>1$ with continuous sample paths almost surely, for which $\arg \max_{\boldsymbol{t}\in {\mathcal{A}}} {\mathrm{Var}}(X(\boldsymbol{t}))$ is a Jordan set with a finite and positive Lebesgue measure of dimension $k\le n$ and its dependence structure is not necessarily locally stationary. Our findings are applied to derive the asymptotics of tail probabilities related to performance tables and chi processes, particularly when the covariance structure is not locally stationary.

We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^{d}$ for every activity $\lambda < ({e}^{L} \hat{C}_{\phi})^{-1}$, where L is the local stability constant and $\hat{C}_{\phi} \,:\!=\, \sup_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - {e}^{-\left\lvert \phi(x, y) \right\rvert} \mathrm{d} y$ is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle–Penrose bound by a factor of at least ${e}$, where the improvements become larger as the negative parts of the potential become more prominent (i.e. for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (2004 Random Structures & Algorithms 24, 461–479): We show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth–death dynamics) that converges rapidly to the finite-volume Gibbs point process while the effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.

A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models (RCMs) on the d-dimensional hyperbolic space, ${\mathbb{H}^d}$, in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on ${\mathbb{H}^d}$ to diagonalize convolution by the adjacency function and the two-point function and bound their $L^2\to L^2$ operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic RCMs. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some RCMs whose resulting graphs are almost surely not locally finite.

We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizontal displacement as well as renewal-process arguments.

In this paper, we investigate a class of McKean–Vlasov stochastic differential equations (SDEs) with Lévy-type perturbations. We first establish the existence and uniqueness theorem for the solutions of the McKean–Vlasov SDEs by utilizing an Eulerlike approximation. Then, under suitable conditions, we demonstrate that the solutions of the McKean–Vlasov SDEs can be approximated by the solutions of the associated averaged McKean–Vlasov SDEs in the sense of mean square convergence. In contrast to existing work, a novel feature of this study is the use of a much weaker condition, locally Lipschitz continuity in the state variables, allowing for possibly superlinearly growing drift, while maintaining linearly growing diffusion and jump coefficients. Therefore, our results apply to a broader class of McKean–Vlasov SDEs.

This paper investigates the asymptotic properties of parameter estimation for the Ewens–Pitman partition with parameters $0\lt\alpha\lt1$ and $\theta\gt-\alpha$. Specifically, we show that the maximum-likelihood estimator (MLE) of $\alpha$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normal distributions, where the variance is governed by the Mittag-Leffler distribution. Moreover, we show that a proper normalization involving a random statistic eliminates the randomness in the variance. Building on this result, we construct an approximate confidence interval for $\alpha$. Our proof relies on a stable martingale central limit theorem, which is of independent interest.

Asymptotic properties of random graph sequences, like the occurrence of a giant component or full connectivity in Erdös–Rényi graphs, are usually derived with very specific choices for the defining parameters. The question arises as to what extent those parameter choices may be perturbed without losing the asymptotic property. For two sequences of graph distributions, asymptotic equivalence (convergence in total variation) and contiguity have been considered by Janson (2010) and others; here we use so-called remote contiguity to show that connectivity properties are preserved in more heavily perturbed Erdös–Rényi graphs. The techniques we demonstrate here with random graphs also extend to general asymptotic properties, e.g. in more complex large-graph limits, scaling limits, large-sample limits, etc.

The results on Γ-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the Γ-limit, which is then used to study deterministic and stochastic homogenization problems for this type of functional.

Exchangeable partitions of the integers and their corresponding mass partitions on $\mathcal{P}_{\infty} = \{\mathbf{s} = (s_{1},s_{2},\ldots)\colon s_{1} \geq s_{2} \geq \cdots \geq 0$ and $\sum_{k=1}^{\infty}s_{k} = 1\}$ play a vital role in combinatorial stochastic processes and their applications. In this work, we continue our focus on the class of Gibbs partitions of the integers and the corresponding stable Poisson–Kingman-distributed mass partitions generated by the normalized jumps of a stable subordinator with an index $\alpha \in (0,1)$, subject to further mixing. This remarkable class of infinitely exchangeable random partitions is characterized by probabilities that have Gibbs (product) form. These partitions have practical applications in combinatorial stochastic processes, random tree/graph growth models, and Bayesian statistics. The most notable class consists of random partitions generated from the two-parameter Poisson–Dirichlet distribution $\mathrm{PD}(\alpha,\theta)$. While the utility of Gibbs partitions is recognized, there is limited understanding of the broader class. Here, as a continuation of our work, we address this gap by extending the dual coagulation/fragmentation results of Pitman (1999), developed for the Poisson–Dirichlet family, to all Gibbs models and their corresponding Poisson–Kingman mass partitions, creating nested families of Gibbs partitions and mass partitions. We focus primarily on fragmentation operations, identifying which classes correspond to these operations and providing significant calculations for the resulting Gibbs partitions. Furthermore, for completion, we provide definitive results for dual coagulation operations using dependent processes. We demonstrate the applicability of our results by establishing new findings for Brownian excursion partitions, Mittag-Leffler, and size-biased generalized gamma models.

In this paper, we prove that the hitting probability of the Minkowski sum of fractal percolations can be characterised by capacity. Then we extend this result to Minkowski sums of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

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.

Sharp, nonasymptotic bounds are obtained for the relative entropy between the distributions of sampling with and without replacement from an urn with balls of $c\geq 2$ colors. Our bounds are asymptotically tight in certain regimes and, unlike previous results, they depend on the number of balls of each color in the urn. The connection of these results with finite de Finetti-style theorems is explored, and it is observed that a sampling bound due to Stam (1978) combined with the convexity of relative entropy yield a new finite de Finetti bound in relative entropy, which achieves the optimal asymptotic convergence rate.

We study a two-dimensional discounted optimal stopping zero-sum (or Dynkin) game related to perpetual redeemable convertible bonds expressed as game (or Israeli) options in a model of financial markets in which the behaviour of the ex-dividend price of a dividend-paying asset follows a generalized geometric Brownian motion. It is assumed that the dynamics of the random dividend rate of the asset paid to shareholders are described by the mean-reverting filtering estimate of an unobservable continuous-time Markov chain with two states. It is shown that the optimal exercise (conversion) and withdrawal (redemption) times forming a Nash equilibrium are the first times at which the asset price hits either lower or upper stochastic boundaries being monotone functions of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundaries are regular for the stopping region relative to the resulting two-dimensional diffusion process and that the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the optimal stopping boundaries are determined as a unique solution to the associated coupled system of nonlinear Fredholm integral equations among the couples of continuous functions of bounded variation satisfying certain conditions. We also give a closed-form solution to the appropriate optimal stopping zero-sum game in the corresponding model with an observable continuous-time Markov chain.

In this article, we focus on the systemic expected shortfall and marginal expected shortfall in a multivariate continuous-time risk model with a general càdlàg process. Additionally, we conduct our study under a mild moment condition that is easily satisfied when the general càdlàg process is determined by some important investment return processes. In the presence of heavy tails, we derive asymptotic formulas for the systemic expected shortfall and marginal expected shortfall under the framework that includes wide dependence structures among losses, covering pairwise strong quasi-asymptotic independence and multivariate regular variation. Our results quantify how the general càdlàg process, heavy-tailed property of losses, and dependence structures influence the systemic expected shortfall and marginal expected shortfall. To discuss the interplay of dependence structures and heavy-tailedness, we apply an explicit order 3.0 weak scheme to estimate the expectations related to the general càdlàg process. This enables us to validate the moment condition from a numerical perspective and perform numerical studies. Our numerical studies reveal that the asymptotic dependence structure has a significant impact on the systemic expected shortfall and marginal expected shortfall, but heavy-tailedness has a more pronounced effect than the asymptotic dependence structure.

The gambler’s ruin problem for correlated random walks (CRWs), both with and without delays, is addressed using the optional stopping theorem for martingales. We derive closed-form expressions for the ruin probabilities and the expected game duration for CRWs with increments $\{1,-1\}$ and for symmetric CRWs with increments $\{1,0,-1\}$ (CRWs with delays). Additionally, a martingale technique is developed for general CRWs with delays. The gambler’s ruin probability for a game involving bets on two arbitrary patterns is also examined.

In this paper we study the optimal multiple stopping problem with weak regularity for the reward, where the reward is given by a set of random variables indexed by stopping times. When the reward family is upper semicontinuous in expectation along stopping times, we construct the optimal multiple stopping strategy using the auxiliary optimal single stopping problems. We also obtain the corresponding results when the reward is given by a progressively measurable process.

We prove a Poisson process approximation result for stabilising functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing observation window that the process of points with a large nearest neighbour distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbour distance in the Ginibre process, which turns out to be different from its analogue for independent points.

The scale function plays a significant role in the fluctuation theory of Lévy processes, particularly in addressing exit problems. However, its definition is established through the Laplace transform, which generally lacks an explicit representation. This paper introduces a novel series representation for the scale function, utilizing Laguerre polynomials to construct a uniformly convergent approximation sequence. Additionally, we conduct statistical inference based on specific discrete observations and propose estimators for the scale function that are asymptotically normal.