The explicit expression for the Kolmogorov constant given in Theorem 1.2 of [1] has a restricted range of validity: it gives the constant precisely for linear-fractional offspring generating functions, whereas for other subcritical offspring generating functions the constant may be strictly larger or strictly smaller than that expression. Within the class of embeddable offspring generating functions the expression is a sharp lower bound for the constant.

Let $\{U(n)\}_{n \geq 0}$ be a sequence of independent random variables such that U(n) is distributed uniformly on $\{0, 1, 2, \dots, n\}$. The Ulam–Kac adder is the history-dependent random sequence defined by $X_{n + 1} = X_{n} + X_{U(n)}$ with the initial condition $X_0 = 1$. We show that for each $m \geq 1$, $\log \mathbb{E}[X_n^m]/\sqrt{n}$ approaches a constant $c_m$ as $n \to \infty$. For a given m, a method is provided which allows the high-accuracy numerical computation of $c_m$. For arbitrary m, loose bounds are provided.

We establish connections between the absorption probabilities of a class of birth–death processes with killing, and the stationary tail of a related class of birth–death processes with catastrophes. The major ingredients of the proofs are a decomposition of the dynamics of these processes, a Feynman–Kac type relationship for Markov chains with reset and rebirth, and the concept of Siegmund duality, which allows us to invert the relationship between the processes. We apply our results to a pair of ancestral processes in population genetics, namely the killed ancestral selection graph and the pruned lookdown ancestral selection graph, in a finite population setting and its diffusion limit.

We study a population of N individuals evolving according to a biparental Moran model with two types, one being advantaged compared to the other. The advantage is conferred by a Mendelian mutation, which reduces the death probability of individuals carrying it. We assume that a proportion a of individuals initially carry this mutation, which therefore eventually gets fixed with high probability. After a long time, we sample a gene uniformly from the population, at a new locus, independent of the locus under selection, and calculate the probability that this gene originated from one of the initially advantaged individuals, when the population size is large. Our theorem provides quantitative insights, such as the observation that under strong viability selection, if only 1% of the individuals are initially advantaged, up to 19% of the population’s genome will originate from them after a long time.

We establish a sufficient condition for comparing linear combinations of independent and identically distributed infinite-mean random variables under the usual stochastic order. We introduce a new class of distributions that includes many commonly used heavy-tailed models and show that within this class, a linear combination of random variables is stochastically larger when its weight vector is smaller in the sense of majorization order. We proceed to study the case where each random variable is a compound Poisson sum and demonstrate that if the stochastic dominance relation holds, the summand of the compound Poisson sum belongs to our new class of distributions. Additional discussions are presented for stable distributions.

Nerman’s martingale plays a central role in the law of large numbers for both single- and multi-type supercritical general branching processes. There are further, complex-valued Nerman-type martingales in the single-type process that figure in the finer fluctuations of these processes. We construct the analogous martingales for the process with finitely many types and give sufficient conditions for these martingales to converge in $L^q$ for $q \in (1,2]$.

The stochastic ordering of order statistics plays a central role in many fields involving probability and statistics. Over the past decade, significant progress has been made in ordering the first and second order statistics from heterogeneous or statistically dependent observations. This paper further develops necessary and sufficient conditions for the third-largest and third-smallest order statistics from observations linked by an Archimedean copula and following, respectively, the proportional hazards, proportional reverse hazards, and scale models. The characterization results and sufficient conditions presented herein not only enrich the existing literature but also provide novel theoretical insights for applications in fields such as reliability systems, multiple outlier models, and the design of third-price auction mechanisms.

We consider a random right-angled Coxeter group (RACG), $W_{\Gamma}$, whose presentation graph $\Gamma$ is taken to be an Erdős–Rényi random graph, i.e. $\Gamma\sim\mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of this random group model. Resolving a conjecture of Susse, we determine the connectivity threshold for square percolation on the random graph $\Gamma\sim\mathcal{G}_{n,p}$. We use this result to determine a large range of p for which the random RACG $W_{\Gamma}$ has a unique cubical coarse median structure. Until Fioravanti et al. (2026), there were no non-hyperbolic examples of groups with cubical coarse rigidity. Perhaps surprisingly, we show that the property of coarse cubical rigidity is in fact typically satisfied by a random RACG for a wide range of the parameter p, including $p=\frac12$, for which the random RACG is also typically non-hyperbolic.

The join-the-shortest-queue (JSQ) policy is among the most widely used load-balancing algorithms and has been extensively studied. However, an exact characterization of the system behavior remains challenging. Most prior research has focused on analyzing its performance in the steady state in certain asymptotic regimes, such as the heavy-traffic regime. However, convergence to the steady state in these regimes is often slow, so steady-state and heavy-traffic characterizations may be less informative over practical time horizons. To address this limitation, we provide a finite-time convergence rate analysis of a JSQ system with two symmetric servers. In sharp contrast to the existing literature, we directly study the original system rather than an approximate limiting system such as a diffusion approximation. Our results demonstrate that for such a system, the convergence rate to its steady state, measured in the total variation distance, is $O((1/(1-\rho)^3)(1/t))$, where $\rho \in (0,1)$ is the traffic intensity.

Introduced by Müller et al. in their seminal paper (2017), fractional stochastic dominance (SD) offers a nuanced approach to ordering distributions. In this paper we propose a fundamentally new framework by replacing the fixed parameter $\gamma \in [0, 1]$ in fractional SD with a function $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$. This yields two novel families, multi-fractional stochastic dominance (MFSD) and functional fractional stochastic dominance (FFSD). They enable the ranking of a broader range of distributions and incorporate a more informative utility class, including those with local non-concavities whose steepness varies depending on the location. Furthermore, our framework introduces the concept of partial greediness, which dynamically captures how behaviour of decision makers adapts to changes in wealth. We also extend this framework to encompass almost stochastic dominance. We provide the mathematical foundations of our generalized framework and study how it offers a novel tool for ordering distributions across various settings.

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time-stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erdős–Rényi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of ‘soft’ random geometric graphs as well as the standard random geometric graph.

We introduce a new permutation statistic, which we call the k-window statistic, for permutations of $\{1,2,\ldots, n\}$. The aim is to add to the repertoire of potentially useful tools in the design of efficient algorithms. The k-window statistic counts the number of elements that are at a displacement at most k from their natural positions in the ascendingly sorted permutation. We find the exact mean and exact variance of the k-window statistic, revealing phases in the mean and variance in the different regimes of k. We find a weak (or even strong) law of large numbers in the regime where $k=k_n$ is growing but remains ${\mathrm{o}} (n)$. Via a representation of the statistic into dependent heterogeneous weakly correlated indicators, we establish the complete spectrum of distributions as $k_n$ spans its entire range using an appropriate method for each regime.

Neural networks have been widely used in data analysis, but there is relatively little research on the statistical theory of neural networks. In this paper we generalize the scale regularization method to the k-degree neural networks, and obtain a statistical guarantee for estimators with a least-squares term and a regularizer. On this basis, we exemplify this guarantee with locally Lipschitz activation functions and $l_{\beta}$-regularization for $\beta\in[1,2]$ in the case $k>1$, showing that the corresponding prediction error decrease is of the order of $n^{-{{1}/{2}}}$ in the sample size n.

In this work we study a continuous time exponential utility maximization problem in the presence of a linear temporary price impact. More precisely, for the case where the risky asset is given by the Ornstein–Uhlenbeck diffusion process, we compute the optimal portfolio strategy and the corresponding value. Our method of solution relies on duality, and it is purely probabilistic.

Let $\alpha_n({\cdot})=\mathbb{P}\bigl(X_{n+1}\in{\cdot}\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of p-dimensional random vectors. Suppose $\alpha_n=\mathcal{N}(M_n,Q_n)$, where $M_n=({1}/{n})\sum_{i=1}^nX_i$ and $Q_n=({1}/{n})\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^{\top}$. Then there is a random probability measure $\alpha$ on the Borel subsets of $\mathbb{R}^p$ such that $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$, where $\lVert{\cdot}\rVert$ is the total variation distance. An explicit expression for $\alpha$ is provided and the convergence rate of $\lVert\alpha_n-\alpha\rVert$ is shown to be arbitrarily close to $n^{-1/2}$. Moreover, it is still true that $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ even if $\alpha_n=\mathcal{L}(M_n,Q_n)$, where $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives $\alpha_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions is investigated and a numerical experiment is performed.

We derive a complete left-tail asymptotic series for the density of the martingale limit of a Galton–Watson process with immigration. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of branching processes with immigration. A good, quickly computed approximation for the density will also be derived from the series.

A Markovian modulation captures market trends and influences market coefficients accordingly. The various scenarios presented by the market are modeled as distinct states of a discrete-time Markov chain. In this paper we assume the existence of such modulation in a market and, as a novelty, propose that it can be anticipative with respect to the future of the Brownian motion driving the dynamics of the risky asset. Using techniques from the enlargement of filtrations, we solve an optimal portfolio utility problem in both complete and incomplete markets. Several examples of anticipative Markov chains are provided, for which we calculate the additional gains available to an investor with more accurate information.

In this paper, we study generalized versions of Poisson-type processes obtained by time-changing them with the geometric counting process (GCP). Our work generalizes the results of Meoli (2023). We introduce the geometric time-changed Poisson processes of Type I (GTCPP-I) and Type II (GTCPP-II), the geometric generalized compound, and multiplicative Poisson processes for both types. We derive several distributional properties for the above-mentioned processes and study special cases, including the geometric space-fractional Poisson process, the geometric tempered space-fractional Poisson process, and the geometric time-fractional Poisson process. We also analyze the asymptotic behavior of the correlation structure. Finally, we discuss applications in shock models in reliability theory, deriving explicit reliability functions and failure rates under both the extreme shock model and the cumulative shock model.

We introduce a fractional Bessel process with constant drift, defined as a time-changed Bessel process via the inverse of a stable subordinator, independent of the base process. This construction yields a model capable of capturing subdiffusive behavior and long-range dependence, relevant in various complex systems. We derive an explicit spectral representation of its transition density, extending the non-fractional setting. Using this representation, we establish several analytical properties of the process, including its stationary distribution and correlation structure. These results provide new insights into the behavior of fractional diffusions and offer analytical tools for applications in queueing theory, mathematical finance, and related domains. In particular, we demonstrate their applicability through a concrete problem in queueing theory.