In this paper we study one-sided hypothesis testing under random sampling without replacement, which frequently appears in the cryptographic problem setting, including the verification of measurement-based quantum computation. Suppose that $n+1$ binary random variables $X_1,\ldots, X_{n+1}$ follow a permutation invariant distribution and n binary random variables $X_1,\ldots, X_{n}$ are observed. Then, we propose randomized tests with a randomization parameter for the expectation of the $(n+1)$th random variable $X_{n+1}$ under a given significance level $\delta>0$. Our randomized tests significantly improve the upper confidence limit over deterministic tests. Our problem setting commonly appears in machine learning in addition to cryptographic scenarios by considering adversarial examples. Such studies are essential for expanding the applicable area of statistics. Although this paper addresses only binary random variables, a similar significant improvement by randomized tests can be expected for general non-binary random variables.

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.

In this paper we consider a dynamic Erdős–Rényi graph in which edges, according to an alternating renewal process, change from present to absent and vice versa. The objective is to estimate the on- and off-time distributions while only observing the aggregate number of edges. This inverse problem is dealt with, in a parametric context, by setting up an estimator based on the method of moments. We provide conditions under which the estimator is asymptotically normal, and we point out how the corresponding covariance matrix can be identified. We also demonstrate how to adapt the estimation procedure if alternative subgraph counts are observed, such as the number of wedges or triangles.

In this paper, we introduce a unified framework based on the pathwise expansion method to derive explicit recursive formulas for cumulative distribution functions, option prices, and transition densities in multivariate diffusion models. A key innovation of our approach is the introduction of the quasi-Lamperti transform, which normalizes the diffusion matrix at the initial time. This transformation facilitates expansions using uncorrelated Brownian motions, effectively reducing multivariate problems to one-dimensional computations. Consequently, both the analysis and the computation are significantly simplified. We also present two novel applications of the pathwise expansion method. Specifically, we employ the proposed framework to compute the value-at-risk for stock portfolios and to evaluate complex derivatives, such as forward-starting options. Our method has the flexibility to accommodate models with diverse features, including stochastic risk premiums, stochastic volatility, and nonaffine structures. Numerical experiments demonstrate the accuracy and computational efficiency of our approach. In addition, as a theoretical contribution, we establish an equivalence between the pathwise expansion method and the Hermite polynomial-based expansion method in the literature.

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 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 $\alpha \leq \tfrac{1}{2}$.

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.

In this paper, we consider a bidimensional risk model with stochastic returns and dependent subexponential claims, in which every main claim may be accompanied by a delayed claim, occurring after an uncertain period of time. The surplus of each business line is allowed to be invested in a portfolio of risk-free assets, and the price process of the investment is modeled by a geometric Lévy process. Meanwhile, we employ a time-claim-dependent structure to describe the dependence among claims and the interarrival times. Some uniform asymptotic formulas for the finite-time ruin probabilities are derived under this structure. Finally, a simulation study is conducted to evaluate the accuracy of the derived results.

This paper investigates a continuous-time multidimensional risk model with stochastic returns driven by a geometric Lévy process, where each main claim is accompanied by a random number of delayed claims. By employing a framework of multivariate regular variation for claim sizes and allowing for arbitrarily dependent claim-number processes, we conduct asymptotic analyses for two types of ruin probabilities. Numerical examples are used to demonstrate the accuracy of our asymptotic estimates.

We show that the Potts model on a graph can be approximated by a sequence of independent and identically distributed spins in terms of Wasserstein distance at high temperatures. We prove a similar result for the Curie–Weiss–Potts model on the complete graph, conditioned on being close enough to any of its equilibrium macrostates, in the low-temperature regime. Our proof technique is based on Stein’s method for comparing the stationary distributions of two Glauber dynamics with similar updates, one of which is rapid mixing and contracting on a subset of the state space. Along the way, we prove a new upper bound on the mixing time of the Glauber dynamics for the conditional measure of the Curie–Weiss–Potts model near an equilibrium macrostate.

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.