We consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds by locally comparing these edge statistics with those for a random tiling of a hexagon, which are well understood. To realize this comparison, we require a nearly optimal concentration estimate for the tiling height function, which we establish by exhibiting a certain Markov chain on the set of all tilings that preserves such concentration estimates under its dynamics.

We consider interacting urns on a finite directed network, where both sampling and reinforcement processes depend on the nodes of the network. This extends previous research by incorporating node-dependent sampling and reinforcement. We classify the sampling and reinforcement schemes, as well as the networks on which the proportion of balls of either colour in each urn converges almost surely to a deterministic limit. We also investigate conditions for achieving synchronisation of the colour proportions across the urns and analyse fluctuations under specific conditions on the reinforcement scheme and network structure.

Let $\{\omega _n\}_{n\geq 1}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega , \mathcal {F}, \mathbb {P})$, each uniformly distributed on the unit circle $\mathbb {T}$, and let $\ell _n=cn^{-\tau }$ for some $c>0$ and $0<\tau <1$. Let $I_{n}=(\omega _n,\omega _n+\ell _n)$ be the random interval with left endpoint $\omega _n$ and length $\ell _n$. We study the asymptotic property of the covering time $N_n(x)=\sharp \{1\leq k\leq n: x\in I_k\}$ for each $x\in \mathbb {T}$. We prove the quenched central limit theorem for the covering time, that is, $\mathbb {P}$-almost surely,

$$ \begin{align*}\frac{N_n(x)-\mathbb{E}_{\mathbb{P}}(N_n(x))}{\sqrt{\sum_{k=1}^n \ell_k(1-\ell_k)}}\end{align*} $$

converges in law to the standard normal distribution.

Gut and Stadmüller (2021, 2022) initiated the study of the elephant random walk with limited memory. Aguech and El Machkouri (2024) published a paper in which they discuss an extension of the results by Gut and Stadtmüller (2022) for an ‘increasing memory’ version of the elephant random walk without stops. Here we present a formal definition of the process that was hinted at by Gut and Stadtmüller. This definition is based on the triangular array setting. We give a positive answer to the open problem in Gut and Stadtmüller (2022) for the elephant random walk, possibly with stops. We also obtain the central limit theorem for the supercritical case of this model.

Early investigation of Pólya urns considered drawing balls one at a time. In the last two decades, several authors have considered multiple drawing in each step, but mostly for schemes involving two colors. In this manuscript, we consider multiple drawing from urns of balls of multiple colors, formulating asymptotic theory for specific urn classes and addressing more applications. The class we consider is affine and tenable, built around a ‘core’ square matrix. We examine cases where the urn is irreducible and demonstrate its relationship to matrix irreducibility for its core matrix, with examples provided. An index for the drawing schema is derived from the eigenvalues of the core. We identify three regimes: small, critical, and large index. In the small-index regime, we find an asymptotic Gaussian law. In the critical-index regime, we also find an asymptotic Gaussian law, albeit with a difference in the scale factor, which involves logarithmic terms. In both of these regimes, we have explicit forms for the structure of the mean and the covariance matrix of the composition vector (both exact and asymptotic). In all three regimes we have strong laws.

We establish a Central Limit Theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb {N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaran et al. (Prob. Theory Relat. Fields 167, 2017) and Hiraoka et al. (Ann. Appl. Prob. 28, 2018).

In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich, Ann. Appl. Prob. 11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r, s), $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.

We introduce a modification of the generalized Pólya urn model containing two urns, and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$ added to the urns after n draws, where $J\in\mathbb{N}$. We provide sufficient conditions under which the random variables $(B_j(n))_{n\in\mathbb{N}}$, properly normalized and centered, converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second-order behavior occurs at $\sqrt{\rho}$ and not at $\rho/2$, where $\rho$ is the dominant eigenvalue of the mean replacement matrix.

In this article, we give explicit bounds on the Wasserstein and Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard normal distribution, using the results of Last et al. (Prob. Theory Relat. Fields 165, 2016). Relying on the theory developed by Saulis and Statulevicius in Limit Theorems for Large Deviations (Kluwer, 1991) and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results of Hillairet et al. (ALEA 19, 2022) and Khabou et al. (J. Theoret. Prob. 37, 2024) regarding the normal approximation and those of Zhu (Statist. Prob. Lett. 83, 2013) for moderate deviations.

We use Stein’s method to obtain distributional approximations of subgraph counts in the uniform attachment model or random directed acyclic graph; we provide also estimates of rates of convergence. In particular, we give uni- and multi-variate Poisson approximations to the counts of cycles and normal approximations to the counts of unicyclic subgraphs; we also give a partial result for the counts of trees. We further find a class of multicyclic graphs whose subgraph counts are a.s. bounded as $n\to \infty$.

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

We propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations which characterize nonlinear $\alpha$-stable Lévy processes under a sublinear expectation space with $\alpha\in(1,2)$. We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry–Esseen bound and convergence rate for the $\alpha$-stable central limit theorem under sublinear expectation.

We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size n according to the decreasing order of their degrees; namely, let $(v^{(i)})_{i=1}^{n}$ be such that $\deg(v^{(1)}) \geq \cdots \geq \deg (v^{(n)})$. The targeted vertex-cutting process is performed by sequentially removing vertices $v^{(1)}, v^{(2)}, \ldots, v^{(n)}$ and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, $K_n$, is upper bounded by $Z_{\geq D}$, which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that $\ln Z_{\geq D}$ grows as $\ln n$ asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the kth moment of $\ln Z_{\geq D}$ is proportional to $(\!\ln n)^k$. As a consequence, we obtain that the first-order growth of $K_n$ is upper bounded by $n^{1-\ln 2}$, which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random.

The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with limit theorems of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We first show that the law of large numbers of the occupation time process is driven by the identity function when the dimension of the lattice is at least 3 and the infection rate of the model is sufficiently large conditioned on the initial state of the NBCPP being distributed with a particular invariant distribution. Then we show that the centered occupation time process of the NBCPP converges in finite-dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with the aforementioned invariant distribution.

We study the noise sensitivity of the minimum spanning tree (MST) of the $n$-vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by $n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov–Hausdorff–Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability $\varepsilon \gg n^{-1/3}$, then the pair of rescaled minimum spanning trees – before and after the noise – converges in distribution to independent random spaces. Conversely, if $\varepsilon \ll n^{-1/3}$, the GHP distance between the rescaled trees goes to $0$ in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of $n^{-1/3}$ coincides with the critical window of the Erdős-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.

As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.

We investigate branching processes in varying environment, for which $\overline{f}_n \to 1$ and $\sum_{n=1}^\infty (1-\overline{f}_n)_+ = \infty$, $\sum_{n=1}^\infty (\overline{f}_n - 1)_+ < \infty$, where $\overline{f}_n$ stands for the offspring mean in generation n. Since subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial limit we consider two scenarios: conditioning on nonextinction, and adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.

The asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$.

We consider estimation of the spot volatility in a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices. Based on discrete, noisy observations of an Itô semimartingale with jumps and general stochastic volatility, we present a simple and explicit estimator using local order statistics. We establish consistency and stable central limit theorems as asymptotic properties. The asymptotic analysis builds upon an expansion of tail probabilities for the order statistics based on a generalized arcsine law. In order to use the involved distribution of local order statistics for a bias correction, an efficient numerical algorithm is developed. We demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation.