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We consider stationary configurations of points in Euclidean space that are marked by positive random variables called scores. The scores are allowed to depend on the relative positions of other points and outside sources of randomness. Such models have been thoroughly studied in stochastic geometry, e.g. in the context of random tessellations or random geometric graphs. It turns out that in a neighborhood of a point with an extreme score it is possible to rescale positions and scores of nearby points to obtain a limiting point process, which we call the tail configuration. Under some assumptions on dependence between scores, this local limit determines the global asymptotics for extreme scores within increasing windows in 
$\mathbb{R}^d$. The main result establishes the convergence of rescaled positions and clusters of high scores to a Poisson cluster process, quantifying the idea of the Poisson clumping heuristic by Aldous (1989, in the point process setting). In contrast to the existing results, our framework allows for explicit calculation of essentially all extremal quantities related to the limiting behavior of extremes. We apply our results to models based on (marked) Poisson processes where the scores depend on the distance to the kth nearest neighbor and where scores are allowed to propagate through a random network of points depending on their locations.
We analyse a Markovian SIR epidemic model where individuals either recover naturally or are diagnosed, leading to isolation and potential contact tracing. Our focus is on digital contact tracing via a tracing app, considering both its standalone use and its combination with manual tracing. We prove that as the population size n grows large, the epidemic process converges to a limiting process, which, unlike with typical epidemic models, is not a branching process due to dependencies created by contact tracing. However, by grouping to-be-traced individuals into macro-individuals, we derive a multi-type branching process interpretation, allowing computation of the reproduction number R. This is then converted to an individual reproduction number 
$R^\mathrm{(ind)}$, which, in contrast to R, decays monotonically with the fraction of app-users, while both share the same threshold at 1. Finally, we compare digital (only) contact tracing and manual (only) contact tracing, proving that the critical fraction of app-users, 
$\pi_{\mathrm{c}}$, required for 
$R=1$ is higher than the critical fraction manually contact-traced, 
$p_{\mathrm{c}}$, for manual tracing.
For a spectrally negative Lévy process X, consider 
$g_t$ and its infinitesimal generator. Moreover, with 
$t\geq 0$, the last time X is below the level zero before time 
$\{(g_t,t, X_t), t\geq 0 \}$ the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of 
$U_t\,:\!=\,t-g_t$. We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process 
$ (U, X)=\{(U_t, X_t),t\geq 0\}$ in terms of the positive and negative excursions of the process X. As a corollary, we find the joint Laplace transform of 
$(U_{\mathbf{e}_q}, X_{\mathbf{e}_q})$, where 
$\mathbf{e}_q$ is an independent exponential time, and the q-potential measure of the process (U, X). Furthermore, using the results mentioned above, we find a solution to a general optimal stopping problem depending on (U, X) with an application in corporate bankruptcy. Lastly, we establish a link between the optimal prediction of 
$g_{\infty}$ and optimal stopping problems in terms of (U, X) as per Baurdoux, E. J. and Pedraza, J. M., 
$L_p$ optimal prediction of the last zero of a spectrally negative Lévy process, Annals of Applied Probability, 34 (2024), 1350–1402.
The Wright–Fisher model, originating in Wright (1931) is one of the canonical probabilistic models used in mathematical population genetics to study how genetic type frequencies evolve in time. In this paper we bound the rate of convergence of the stationary distribution for a finite population Wright–Fisher Markov chain with parent-independent mutation to the Dirichlet distribution. Our result improves the rate of convergence established in Gan et al. (2017) from 
$\mathrm{O}(1/\sqrt{N})$ to 
$\mathrm{O}(1/N)$. The results are derived using Stein’s method, in particular, the prelimit generator comparison method.
In this paper, we study asymptotic behaviors of a subcritical branching Brownian motion with drift 
$-\rho$, killed upon exiting 
$(0, \infty)$, and offspring distribution 
$\{p_k{:}\; k\ge 0\}$. Let 
$\widetilde{\zeta}^{-\rho}$ be the extinction time of this subcritical branching killed Brownian motion, 
$\widetilde{M}_t^{-\rho}$ the maximal position of all the particles alive at time t and 
$\widetilde{M}^{-\rho}:\!=\max_{t\ge 0}\widetilde{M}_t^{-\rho}$ the all-time maximal position. Let 
$\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at 
$x\in (0,\infty)$. Under the assumption 
$\sum_{k=1}^\infty k ({\log}\; k) p_k <\infty$, we establish the decay rates of 
$\mathbb{P}_x(\widetilde{\zeta}^{-\rho}>t)$ and 
$\mathbb{P}_x(\widetilde{M}^{-\rho}>y)$ as t and y respectively tend to 
$\infty$. We also establish the decay rate of 
$\mathbb{P}_x(\widetilde{M}_t^{-\rho}> z(t,\rho))$ as 
$t\to\infty$, where 
$z(t,\rho)=\sqrt{t}z-\rho t$ for 
$\rho\leq 0$ and 
$z(t,\rho)=z$ for 
$\rho>0$. As a consequence, we obtain a Yaglom-type limit theorem.
In this paper, we study the asymptotic behavior of the generalized Zagreb indices of the classical Erdős–Rényi (ER) random graph G(n, p), as 
$n\to\infty$. For any integer 
$k\ge1$, we first give an expression for the kth-order generalized Zagreb index in terms of the number of star graphs of various sizes in any simple graph. The explicit formulas for the first two moments of the generalized Zagreb indices of an ER random graph are then obtained from this expression. Based on the asymptotic normality of the numbers of star graphs of various sizes, several joint limit laws are established for a finite number of generalized Zagreb indices with a phase transition for p in different regimes. Finally, we provide a necessary and sufficient condition for any single generalized Zagreb index of G(n, p) to be asymptotic normal.
The payoff in the Chow–Robbins coin-tossing game is the proportion of heads when you stop. Stopping to maximize expectation was addressed by Chow and Robbins (1965), who proved there exist integers 
${k_n}$ such that it is optimal to stop at n tosses when heads minus tails is 
${k_n}$. Finding 
${k_n}$ was unsolved except for finitely many cases by computer. We prove an 
$o(n^{-1/4})$ estimate of the stopping boundary of Dvoretsky (1967), which then proves 
${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta (\! -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil $ except for n in a set of density asymptotic to 0, at a power law rate. Here, 
$\alpha$ is the Shepp–Walker constant from the Brownian motion analog, and 
$\zeta$ is Riemann’s zeta function. An 
$n^{ - 1/4}$ dependence was conjectured by Christensen and Fischer (2022). Our proof uses moments involving Catalan and Shapiro Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund (2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod’s embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in another way. We use them first for yet many more examples and a conjecture, then algebraically in the tree, with feedback to get much sharper Value bounds near the border, and analytic results. Also, we give a formula that gives the exact optimal stop rule for all n up to about a third of a billion; it uses the analytic result plus terms arrived at empirically.
We study continuous-time Markov chains on the nonnegative integers under mild regularity conditions (in particular, the set of jump vectors is finite and both forward and backward jumps are possible). Based on the so-called flux balance equation, we derive an iterative formula for calculating stationary measures. Specifically, a stationary measure 
$\pi(x)$ evaluated at 
$x\in\mathbb{N}_0$ is represented as a linear combination of a few generating terms, similarly to the characterization of a stationary measure of a birth–death process, where there is only one generating term, 
$\pi(0)$. The coefficients of the linear combination are recursively determined in terms of the transition rates of the Markov chain. For the class of Markov chains we consider, there is always at least one stationary measure (up to a scaling constant). We give various results pertaining to uniqueness and nonuniqueness of stationary measures, and show that the dimension of the linear space of signed invariant measures is at most the number of generating terms. A minimization problem is constructed in order to compute stationary measures numerically. Moreover, a heuristic linear approximation scheme is suggested for the same purpose by first approximating the generating terms. The correctness of the linear approximation scheme is justified in some special cases. Furthermore, a decomposition of the state space into different types of states (open and closed irreducible classes, and trapping, escaping and neutral states) is presented. Applications to stochastic reaction networks are well illustrated.
Let 
$(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation where 
\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]
$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, 
$A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, 
$(Z_t)_{t\geqslant 0}$ is a d-dimensional rotationally symmetric 
$\alpha$-stable Lévy process with 
$\alpha \in (1,2)$ and 
$x\in\mathbb{R}^{d}$. We use two Euler–Maruyama schemes with decreasing step sizes 
$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of 
$(X_t)_{t \geqslant 0}$: one uses independent and identically distributed 
$\alpha$-stable random variables as innovations, and the other employs independent and identically distributed Pareto random variables. We study the convergence rates of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, under the assumption that the function b is Lipschitz and satisfies a certain dissipation condition, we demonstrate a convergence rate of 
$\gamma^{\frac{1}{\alpha}}_n$. This convergence rate can be improved to 
$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$ for any 
$\kappa \in [1,\alpha)$, provided b has the additional regularity of bounded second-order directional derivatives. For the second scheme, where the function b is assumed to be twice continuously differentiable, we establish a convergence rate of 
$\gamma^{\frac{2-\alpha}{\alpha}}_n$; moreover, we show that this rate is optimal for the one-dimensional stable Ornstein–Uhlenbeck process. Our theorems indicate that the recent significant result of [34] concerning the unadjusted Langevin algorithm with additive innovations can be extended to stochastic differential equations driven by an 
$\alpha$-stable Lévy process and that the corresponding convergence rate exhibits similar behaviour. Compared with the result in [6], our assumptions have relaxed the second-order differentiability condition, requiring only a Lipschitz condition for the first scheme, which broadens the applicability of our approach.
Let X be the sum of a diffusion process and a Lévy jump process, and for any integer 
$n\ge 1$ let 
$\phi_n$ be a function defined on 
$\mathbb{R}^2$ and taking values in 
$\mathbb{R}$, with adequate properties. We study the convergence of functionals of the type where [x] is the integer part of the real number x and the sequences 
\begin{align*} \Delta_n\sum_{i=1}^{[t/\Delta_n]} \phi_n\!\left(\alpha_n\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right], \:\frac{\alpha_n}{\sqrt{\Delta_n}}\left(\left[\frac{X_{i\Delta_n}}{\alpha_n}\right]-\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right]\right)\right),\end{align*}
$(\Delta_n)$ and 
$(\alpha_n)$ tend to 0 as 
$n\to +\infty$. We then prove the law of large numbers and establish, in the case where 
$\frac{\alpha_n}{\sqrt{\Delta_n}}$ converges to a real number in 
$[0,+\infty)$], a new central limit theorem which generalizes that in the case where X is a continuous Itô’s semimartingale.
Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and then study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, 
$\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.