We prove the first explicit rate of convergence to the Tracy–Widom distribution for the fluctuation of the largest eigenvalue of sample covariance matrices that are not integrable. Our primary focus is matrices of type ${\mathit{X}}^{\ast }\mathit{X}$ and the proof follows the Erdös–Schlein–Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from (J. Eur. Math. Soc. (JEMS) 24 (2022) 2823–2873) to obtain a quantitative error estimate for the local relaxation flow at the edge. Together with a quantitative version of the Green function comparison theorem, this gives the rate of convergence.

Combined with a result of Lee–Schnelli (Ann. Appl. Probab. 26 (2016) 3786–3839), some quantitative estimates also hold for more general separable sample covariance matrices ${\mathit{X}}^{\ast }\mathrm{\Sigma }\mathit{X}$ with general diagonal population Σ.

Stochastic integrals are defined with respect to a collection $\mathit{P}=({\mathit{P}}_{\mathit{i}};\mathit{i}\in \mathit{I})$ of continuous semimartingales, imposing no assumptions on the index set I and the subspace of ${\mathbb{R}}^{\mathit{I}}$ where P takes values. The integrals are constructed though finite-dimensional approximation, identifying the appropriate local geometry that allows extension to infinite dimensions. For local martingale integrators, the resulting space $\mathsf{S}(\mathit{P})$ of stochastic integrals has an operational characterisation via a corresponding set of integrands $\mathsf{R}(\mathit{C})$, constructed with only reference to the covariation structure C of P. This bijection between $\mathsf{R}(\mathit{C})$ and the (closed in the semimartingale topology) set $\mathsf{S}(\mathit{P})$ extends to families of continuous semimartingale integrators for which the drift process of P belongs to $\mathsf{R}(\mathit{C})$. In the context of infinite-asset models in mathematical finance, the latter structural condition is equivalent to a certain natural form of market viability. The enriched class of wealth processes via extended stochastic integrals leads to exact analogues of optional decomposition and hedging duality as the finite-asset case. A corresponding characterisation of market completeness in this setting is provided.

In recent years, there has been a substantive interest in rough volatility models. In this class of models, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter $\mathit{H}<0.5$. In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

In this work, several convergence results are established for nearly critical self-excited systems in which event arrivals are described by multivariate marked Hawkes point processes. Under some mild high-frequency assumptions, the rescaled density process behaves asymptotically like a multi-type continuous-state branching process with immigration, which is the unique solution to a multi-dimensional stochastic differential equation with dynamical mechanism similar to that of multivariate Hawkes processes. To illustrate the strength of these limit results, we further establish diffusion approximations for multi-type Crump–Mode–Jagers branching processes counted with various characteristics by linking them to marked Hawkes shot noise processes. In particular, an interesting phenomenon in queueing theory, well known as state space collapse, is observed in the behavior of the population structure at a large time scale. This phenomenon reveals that the rescaled complex biological system can be recovered from its population process by a lifting map.

Ancestral sequence reconstruction is a key task in computational biology. It consists in inferring a molecular sequence at an ancestral species of a known phylogeny, given descendant sequences at the tip of the tree. In addition to its many biological applications, it has played a key role in elucidating the statistical performance of phylogeny estimation methods. Here we establish a formal connection to another important bioinformatics problem, multiple sequence alignment, where one attempts to best align a collection of molecular sequences under some mismatch penalty score by inserting gaps. Our result is counter-intuitive: we show that perfect pairwise sequence alignment with high probability is possible in principle at arbitrary large evolutionary distances—provided the phylogeny is known and dense enough. We use techniques from ancestral sequence reconstruction in the taxon-rich setting together with the probabilistic analysis of sequence evolution models involving insertions and deletions.

This paper is concerned with the evolution dynamics of local times of a spectrally positive stable process in the spatial direction. The main results state that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, the local times at positive half line are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson random measure whose intensity coincides with the Lévy measure. This helps us to provide not only a simple proof for the Hölder regularity, but also a uniform upper bound for all moments of the Hölder coefficient as well as a maximal inequality for the local times. Moreover, based on this stochastic Volterra equation, we extend the method of duality to establish an exponential-affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra integral equation associated with the Laplace exponent of the stable process.

Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We study the likelihood ratio test and obtain sufficient conditions under which this task is impossible or feasible. We propose $\mathtt{MPAlign}$, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove $\mathtt{MPAlign}$ to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time.11

The study of Gaussian free field level sets on supercritical Galton–Watson trees has been initiated by Abächerli and Sznitman in 2018. By means of entirely different tools, we continue this investigation and generalize their main result on the positivity of the associated percolation critical parameter ${\mathit{h}}_{\ast }$ to the setting of arbitrary supercritical offspring distribution and random conductances. In our setting, this establishes a rigorous proof of the physics literature mantra that positive correlations facilitate percolation when compared to the independent case. Our proof proceeds by constructing the Galton–Watson tree through an exploration via finite random walk trajectories. This exploration of the tree progressively unveils an infinite connected component in the random interlacements set on the tree, which is stable under small quenched noise. Using a Dynkin-type isomorphism theorem, we then infer the strict positivity of the critical parameter ${\mathit{h}}_{\ast }$. As a byproduct, we obtain transience results for the above-mentioned sets.

We consider general exponential random graph models (ergms) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs ${\mathit{F}}_{\mathit{k}}$. Whereas previous work has shown a degeneracy phenomenon in dense ergms, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the naïve mean-field approximation for the partition function of the corresponding Gibbs measures, and in case of “ferromagnetic” models with vanishing edge density show that typical samples resemble a typical Erdős–Rényi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erdős–Rényi graph in the large deviations regime for excess ${\mathit{F}}_{\mathit{k}}$-homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of (Adv. Math. 319 (2017) 313–347) on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner’s generalized Hölder inequality.

We propose a unified approach to several problems in stochastic portfolio theory (SPT), which is a framework for equity markets with a large number d of stocks. Our approach combines open markets, where trading is confined to the top N capitalized stocks as well as the market portfolio consisting of all d assets, with a parametric family of models which we call hybrid Jacobi processes. We provide a detailed analysis of ergodicity, particle collisions, and boundary attainment, and use these results to study the associated financial markets. Their properties include (1) stability of the capital distribution curve and (2) explicit and not artificially leveraged growth optimal strategies. The sub-class of rank Jacobi models are additionally shown to (3) serve as the worst-case model for a robust asymptotic growth problem under model ambiguity and (4) exhibit stability in the large-d limit. Our definition of an open market is a relaxation of existing definitions which is essential to make the analysis tractable.

We consider a long-range percolation graph on ${\mathbb{Z}}^{\mathit{d}}$ where, in addition to the nearest-neighbor edges of ${\mathbb{Z}}^{\mathit{d}}$, distinct $\mathit{x},\mathit{y}\in {\mathbb{Z}}^{\mathit{d}}$ are connected by an edge independently with probability asymptotic to $\mathit{\beta }|\mathit{x}-\mathit{y}{|}^{-\mathit{s}}$, for $\mathit{s}\in (\mathit{d},2\mathit{d})$, $\mathit{\beta }>0$ and $|·|$ a norm on ${\mathbb{R}}^{\mathit{d}}$. We first show that, for all but perhaps a countably many $\mathit{\beta }>0$, the graph-theoretical (a.k.a. chemical) distance between typical vertices at $|·|$-distance r is, with high probability as $\mathit{r}\to \infty$, asymptotic to ${\mathit{\varphi }}_{\mathit{\beta }}(\mathit{r}){(log\mathit{r})}^{\mathrm{\Delta }}$, where ${\mathrm{\Delta }}^{-1}:={log}_2(2\mathit{d}/\mathit{s})$ and ${\mathit{\varphi }}_{\mathit{\beta }}$ is a deterministic, positive, bounded and continuous function subject to log-log-periodicity constraint ${\mathit{\varphi }}_{\mathit{\beta }}({\mathit{r}}^{\mathit{\gamma }})={\mathit{\varphi }}_{\mathit{\beta }}(\mathit{r})$ for $\mathit{\gamma }:=\mathit{s}/(2\mathit{d})$. The proof parallels the arguments developed in a continuum version of the model where a similar scaling was shown earlier by the first author and J. Lin. That work also conjectured that ${\mathit{\varphi }}_{\mathit{\beta }}$ is constant which we show to be false by proving that ${(log\mathit{\beta })}^{\mathrm{\Delta }}{\mathit{\varphi }}_{\mathit{\beta }}$ tends, as $\mathit{\beta }\to \infty$, to a nonconstant limit that is independent of the specifics of the model. The proof reveals arithmetic rigidity of the shortest paths that maintain a hierarchical (dyadic) structure all the way to unit scales.

We analyse the cluster discovered by invasion percolation on a branching process with a power-law offspring distribution. Invasion percolation is a paradigm model of self-organised criticality, where criticality is approach without tuning any parameter. By performing invasion percolation for n steps, and letting $\mathit{n}\to \infty$, we find an infinite subtree, called the invasion percolation cluster (IPC). A notable feature of the IPC is its geometry that consists of a unique path to infinity (also called the backbone) onto which finite forests are attached. The main theorem shows the volume scaling limit of the k-cut IPC, which is the cluster containing the root when the edge between the kth and $(\mathit{k}\mathbf{+}1)$st backbone vertices is cut.

We assume a power-law offspring distribution with exponent α and analyse the IPC for different power-law regimes. In a finite-variance setting $(\mathit{\alpha }>2)$ the results, are a natural extension of previous works on the branching process tree (Electron. J. Probab. 24 (2019) 1–35) and the regular tree (Ann. Probab. 35 (2008) 420–466). However, for an infinite-variance setting ($\mathit{\alpha }\in (1,2)$) or even an infinite-mean setting ($\mathit{\alpha }\in (0,1)$), results significantly change. This is illustrated by the volume scaling of the k-cut IPC, which scales as ${\mathit{k}}^2$ for $\mathit{\alpha }>2$, but as ${\mathit{k}}^{\mathit{\alpha }/(\mathit{\alpha }-1)}$ for $\mathit{\alpha }\in (1,2)$ and exponentially for $\mathit{\alpha }\in (0,1)$.

The excursion set of a ${\mathit{C}}^2$ smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual approximation of the surface area of the excursions does not converge when the lattice becomes dense in the domain of observation to the desired limit. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting factor is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the correction factor, showing that it only depends on the spatial dimension d. This enables us to define an approximation for the surface area of excursion sets for general tessellations of polytopes in ${\mathbb{R}}^{\mathit{d}}$, including Poisson–Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume of excursion sets observed over hypercubic lattices.

We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switching environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some $\mathit{K}\in {\mathbf{N}}^{\ast }$, and the number of infectives vanishes almost surely in finite time. We prove that, as $\mathit{K}\to \infty$, the process composed of the proportions of infectives of each type ${\mathit{X}}^{\mathit{K}}$ and the state of the environment ${\mathrm{\Xi }}^{\mathit{K}}$, converges to a piecewise deterministic Markov process (PDMP) given by a system of randomly switching ODEs. The long term behaviour of this PDMP has been previously investigated by Benaïm and Strickler, and depends only on the sign of the top Lyapunov exponent Λ of the linearised PDMP at 0: if $\mathrm{\Lambda }<0$, the proportion of infectives in each group converges to zero, while if $\mathrm{\Lambda }>0$, the disease becomes endemic. In this paper, we show that the large population asymptotics of ${\mathit{X}}^{\mathit{K}}$ also strongly depend on the sign of Λ: if negative, then from fixed initial proportions of infectives the disease disappears in a time of order at most $log(\mathit{K})$, while if positive, the typical extinction time grows at least as a power of K. We prove that in the situation where the origin is accessible for the linearised PDMP, the mean extinction time of ${\mathit{X}}^{\mathit{K}}$ is logarithmically equivalent to ${\mathit{K}}^{{\mathit{p}}^{\ast }}$, where ${\mathit{p}}^{\ast }>0$ is fully characterised. We also investigate the quasi-stationary distribution ${\mathit{\mu }}^{\mathit{K}}$ of $({\mathit{X}}^{\mathit{K}},{\mathrm{\Xi }}^{\mathit{K}})$ and show that, when $\mathrm{\Lambda }<0$, weak limit points of ${({\mathit{\mu }}^{\mathit{K}})}_{\mathit{K}>0}$ are supported by the extinction set, while when $\mathrm{\Lambda }>0$, limit points belong to the (nonempty) set of stationary distributions of the limiting PDMP which do not give mass to the extinction set.

We study the long time behavior of an underdamped mean-field Langevin (MFL) equation, and provide a general convergence as well as an exponential convergence rate result under different conditions. The results on the MFL equation can be applied to study the convergence of the Hamiltonian gradient descent algorithm for the overparametrized optimization. We then provide some numerical examples of the algorithm to train a generative adversarial network (GAN).

We describe the cluster of large deviations events that arise when one such large deviations event occurs. We work in the framework of an infinite moving average process with a noise that has finite exponential moments.

Using a recently introduced representation of the second order adjoint state as the solution of a function-valued backward stochastic partial differential equation (SPDE), we calculate the viscosity super- and subdifferential of the value function evaluated along an optimal trajectory for controlled semilinear SPDEs. This establishes the well-known connection between Pontryagin’s maximum principle and dynamic programming within the framework of viscosity solutions. As a corollary, we derive that the correction term in the stochastic Hamiltonian arising in nonsmooth stochastic control problems is nonpositive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton–Jacobi–Bellman equations in the framework of viscosity solutions.

The maximum likelihood threshold (MLT) of a graph G is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph.

We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erdős–Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial.

Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erdős–Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4.

We also give a geometric characterization of the MLT of a graph in terms of a new “lifting” problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger–Nelson problem.

For $\ell :{\mathbb{R}}^{\mathit{d}}\to [0,\infty )$ we consider the sequence of probability measures ${({\mathit{\mu }}_{\mathit{n}})}_{\mathit{n}\in \mathbb{N}}$, where ${\mathit{\mu }}_{\mathit{n}}$ is determined by a density that is proportional to $exp(-\mathit{n}\ell )$. We allow for infinitely many global minimal points of ℓ, as long as they form a finite union of compact manifolds. In this scenario, we show estimates for the p-Wasserstein convergence of ${({\mathit{\mu }}_{\mathit{n}})}_{\mathit{n}\in \mathbb{N}}$ to its limit measure. Imposing regularity conditions we obtain a speed of convergence of ${\mathit{n}}^{-1/(2\mathit{p})}$ and adding a further technical assumption, we can improve this to a p-independent rate of $1/2$ for all orders $\mathit{p}\in \mathbb{N}$ of the Wasserstein distance.

In this paper we quantify the rate of convergence in Selberg’s central limit theorem for $log|\mathit{\zeta }(1/2\mathbf{+}\mathit{i}\mathit{t})|$ based on the method of proof given by Radziwiłł and Soundararajan in (Enseign. Math. 63 (2017) 1–19). We achieve the same rate of convergence of ${(logloglog\mathit{T})}^2/\sqrt{loglog\mathit{T}}$ as Selberg in (In Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ (1992) 367–385) in the Kolmogorov distance by using the Dudley distance instead. We also prove the theorem for the multivariate case given by Bourgade in (Probab. Theory Related Fields 148 (2010) 479–500) with the same rate of convergence as in the single variable case.

We fix a major gap in our 2017 paper, along with related issues.