In this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

Let $\mathcal {P}(\mathbf{N})$ be the power set of N. We say that a function $\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$ is an upper density if, for all X, Y ⊆ N and h, k ∈ N+, the following hold: (f1) $\mu ^\ast (\mathbf{N}) = 1$ ; (f2) $\mu ^\ast (X) \le \mu ^\ast (Y)$ if X ⊆ Y; (f3) $\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$ ; (f4) $\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$ , where k · X : = {kx: x ∈ X}; and (f5) $\mu ^\ast (X + h) = \mu ^\ast (X)$ . We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)–(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that $\mu ^\ast (X)\le 1$ for every X ⊆ N. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_{n}$ is singular with probability at most $(2+o(1))^{-n}$ . On the other hand, the best known upper bound on the singularity probability of $M_{n}$ , due to Vershynin (2011), is $2^{-n^{c}}$ , for some unspecified small constant $c>0$ . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_{n}$ is at most $2^{-n^{1/4}\sqrt{\log n}/1000}$ for all sufficiently large $n$ . The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.

We introduce and study skew product Smale endomorphisms over finitely irreducible shifts with countable alphabets. This case is different from the one with finite alphabets and we develop new methods. In the conformal context we prove that almost all conditional measures of equilibrium states of summable Hölder continuous potentials are exact dimensional and their dimension is equal to the ratio of (global) entropy and Lyapunov exponent. We show that the exact dimensionality of conditional measures on fibers implies global exact dimensionality of the original measure. We then study equilibrium states for skew products over expanding Markov–Rényi transformations and settle the question of exact dimensionality of such measures. We apply our results to skew products over the continued fraction transformation. This allows us to extend and improve the Doeblin–Lenstra conjecture on Diophantine approximation coefficients to a larger class of measures and irrational numbers.

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

We consider the asymptotics of the difference between the empirical measures of the β-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik–Kerov–Logan–Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.

We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network of n servers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces across L servers such that any k pieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into the k-shortest queues among the L for which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e. n → ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values in ℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of the n-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-d routeing scheme, also known as the supermarket model, is a special case of the model considered here.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\] , where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$ . Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_{N}$ be a deterministic $N\times N$ matrix, and let $G_{N}$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$ , where $\unicode[STIX]{x1D6FE}>1/2$ . With $L_{N}$ the empirical measure of eigenvalues of ${\mathcal{M}}_{N}$ , we provide a general deterministic equivalence theorem that ties $L_{N}$ to the singular values of $z-M_{N}$ , with $z\in \mathbb{C}$ . We then compute the limit of $L_{N}$ when $M_{N}$ is an upper-triangular Toeplitz matrix of finite symbol: if $M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$ where $\mathfrak{d}$ is fixed, $a_{i}\in \mathbb{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)=\mathbf{1}_{j=i+1}$ , then $L_{N}$ converges, as $N\rightarrow \infty$ , to the law of $\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$ , also of independent and identically distributed entries on the diagonals in $M_{N}$ .

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.

The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimension d=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimension d≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree m≥2. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.

In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.

We show that for $p\geqslant 1$ , the $p$ th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$ th moment provided that $2q$ th and $q$ th integral moments of these variables are comparable for all $q\geqslant 2$ . The latest condition turns out to be necessary in the independent and identically distributed case.

In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

Let $F$ be a non-discrete non-Archimedean locally compact field and ${\mathcal{O}}_{F}$ the ring of integers in $F$ . The main results of this paper are the classification of ergodic probability measures on the space $\text{Mat}(\mathbb{N},F)$ of infinite matrices with entries in $F$ with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$ and the classification, for non-dyadic $F$ , of ergodic probability measures on the space $\text{Sym}(\mathbb{N},F)$ of infinite symmetric matrices with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})$ .

We consider community detection in degree-corrected stochastic block models. We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log(n) or higher. Recovery succeeds even for very heterogeneous degree distributions. The algorithm does not rely on parameters as input. In particular, it does not need to know the number of communities.