We study concentration properties of vertex degrees of n-dimensional Erdős–Rényi random graphs with edge probability \(\rho /n\) by means of high moments of these random variables in the limit when n and \(\rho \) tend to infinity. These moments are asymptotically close to one-variable Bell polynomials \({{\mathcal {B}}}_k(\rho ), k\in {{\mathbb {N}}}\), that represent moments of the Poisson probability

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green’s functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green’s functions inside the

The distribution of the sum of 1-dependent lattice vectors with supports on coordinate axes is approximated by a multivariate compound Poisson distribution and by signed compound Poisson measure. The local and \(\ell _\alpha \)-norms are used to obtain the error bounds. The Heinrich method is used for the proofs.

In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process by \(X=\{X_{t},t\ge 0\}\) and set \(\tau _{l}=\inf \{t\ge 0, X_{t}\le l\}\) and \(\tau _{l,u}=\inf \{t\ge 0, X_{t}\notin (l,u)\}\) with \(l

It is well known that, on a purely algebraic level, a simplified version of the central limit theorem (CLT) can be proved in the framework of a non-commutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both

In this paper, we study a type of stochastic McKean–Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler–Maruyama approximation the existence of its weak solutions is proved. Then we observe the pathwise uniqueness of its weak solutions. Finally, it is shown that the Euler–Maruyama approximation has an optimal strong convergence rate.

In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained

We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements the study by Phetpradap for the random walk range, which is restricted to dimension three and higher, and of van den Berg, Bolthausen and den Hollander, for the volume of the Wiener sausage.

We define and study the three-dimensional windings along Brownian paths in the quaternionic Euclidean, projective and hyperbolic spaces. In particular, the asymptotic laws of these windings are shown to be Gaussian for the flat and spherical geometries while the hyperbolic winding exhibits a different long time-behavior. The corresponding asymptotic law seems to be new and is related to the Cauchy

Consider a truncated circular unitary matrix which is a \(p_n\) by \(p_n\) submatrix of an n by n circular unitary matrix after deleting the last \(n-p_n\) columns and rows. Jiang and Qi (J Theor Probab 30:326–364, 2017) and Gui and Qi (J Math Anal Appl 458:536–554, 2018) study the limiting distributions of the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix

Here, we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and non-smooth domains.

We present sufficient conditions, in terms of the jumping kernels, for two large classes of conservative Markov processes of pure-jump type to be purely discontinuous martingales with finite second moment. As an application, we establish the law of the iterated logarithm for sample paths of the associated processes.

A definition of d-dimensional n-Meixner random vectors is given first. This definition involves the commutators of their semi-quantum operators. After that we focus on the 1-Meixner random vectors and derive a system of d partial differential equations satisfied by their Laplace transform. We provide a set of necessary conditions for this system to be integrable. We use these conditions to give a complete

We prove under general conditions that a trimmed subordinator satisfies a self-standardized central limit theorem (SSCLT). Our basic tool is a powerful distributional approximation result of Zaitsev (Probab Theory Relat Fields 74:535–566, 1987). Among other results, we obtain as special cases of our subordinator result the recent SSCLTs of Ipsen et al. (Stoch Process Appl 130:2228–2249, 2020) for trimmed

For hidden Markov models, one of the most popular estimates of the hidden chain is the Viterbi path—the path maximising the posterior probability. We consider a more general setting, called the pairwise Markov model (PMM), where the joint process consisting of finite-state hidden process and observation process is assumed to be a Markov chain. It has been recently proven that under some conditions

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander’s Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together

Given a sequence \((X_n)\) of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under which the series \(\sum _{n=1}^\infty X_n\) is almost surely convergent. For independent random variables, it is well known that if \(\sum _{n=1}^\infty \mathbb {E}(\Vert X_n\Vert ^2) <\infty \), then \(\sum _{n=1}^\infty X_n\) converges almost

Assume that \(X_{\Sigma } \in \mathbb {R}^{n}\) is a centered random vector following a multivariate normal distribution with positive definite covariance matrix \(\Sigma \). Let \(g : \mathbb {R}^{n} \rightarrow \mathbb {C}\) be measurable and of moderate growth, say \(|g(x)| \lesssim (1 + |x|)^{N}\). We show that the map \(\Sigma \mapsto \mathbb {E}[g(X_{\Sigma })]\) is smooth, and we derive convenient

In this paper, we prove multilevel concentration inequalities for bounded functionals \(f = f(X_1, \ldots , X_n)\) of random variables \(X_1, \ldots , X_n\) that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent

Path decomposition is performed to characterize the law of the pre-/post-supremum, post-infimum and the intermediate processes of a spectrally negative Lévy process taken up to an independent exponential time T. As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre-/post-supremum, post-infimum processes and the intermediate processes are

In this note, we correct claims made in.

In this paper, we show that the mixed fractional Poisson process (MFPP) exhibits the long-range dependence property. It is proved by establishing an asymptotic result for the covariance of inverse mixed stable subordinator. Also, it is shown that the increment process of the MFPP, namely the mixed fractional Poissonian noise, has the short-range dependence property.

We construct a probabilistic model for the number of divisors of a random uniform integer that converges in the mod-Poisson sense to the same limiting function as its original counterpart, the one arising in the Sathé–Selberg theorem. This construction involves a conditioning and gives an alternative perspective to the usual paradigm of “hybrid product” models developed by Gonek, Hughes and Keating

We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.

Let \(\Phi \) be a nuclear space and let \(\Phi '_{\beta }\) denote its strong dual. In this work, we prove the existence of càdlàg versions, the Lévy–Itô decomposition and the Lévy–Khintchine formula for \(\Phi '_{\beta }\)-valued Lévy processes. Moreover, we give a characterization for Lévy measures on \(\Phi '_{\beta }\) and provide conditions for the existence of regular versions to cylindrical

We consider finite-dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution for any fixed time \(t>0\). In addition, we provide Varadhan estimates for the asymptotic behavior of the density for small noise. The emphasis is

This note is concerned with an extension, at second order, of an inequality on the discrete cube \(C_n=\{-\,1,1\}^n\) (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62.

In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and

In this article, we consider a family of real-valued diffusion processes on the time interval [0, 1] indexed by their prescribed initial value \(x \in \mathbb {R}\) and another point in space, \(y \in \mathbb {R}\). We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in y at time \(t=1\). Our main result then concerns the following

We prove limit theorems for cylindrical martingale problems associated with Lévy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal \(\mathrm {SLE}_\kappa ( \kappa -6)\) with \(\kappa =16/3\). Our proof follows the classical excursion construction of \(\mathrm {SLE}_\kappa (\kappa -6)\) processes in the continuum, and we are thus led to introduce suitable cut-off stopping times in order to analyse

We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order \(2+\rho , \rho \in (0,1]\), and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, our result leads to a one-sided moderate deviation principle for martingales. Moreover,

We study the limiting behavior of the k-th eigenvalue \(x_k\) of unitary invariant ensembles with Freud-type and uniform convex potentials. As both k and \(n-k\) tend to infinity, we obtain Gaussian fluctuations for \(x_k\) in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in

We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the asymptotic ratio of the two parameters, the rescaled processes converge toward either a “stationary Brownian motion,” an Ornstein–Uhlenbeck process or a Gaussian white noise

We study the structure of \(n \times n\) random matrices with centered i.i.d. entries having only two finite moments. In the recent joint work with R. Vershynin, we have shown that the operator norm of such matrix A can be reduced to the optimal order \(O(\sqrt{n})\) with high probability by zeroing out a small submatrix of A, but did not describe the structure of this “bad” submatrix nor provide a

Let \(\mathfrak {X}\) be a Hunt process on a locally compact space X such that the set \(\mathcal {E}_{\mathfrak {X}}\) of its Borel measurable excessive functions separates points, every function in \(\mathcal {E}_{\mathfrak {X}}\) is the supremum of its continuous minorants in \({\mathcal {E}}_{{\mathfrak {X}}}\), and there are strictly positive continuous functions \(v,w\in {\mathcal {E}}_{{\mathfrak

The goal of this paper is to obtain expectation bounds for the deviation of large sample autocovariance matrices from their means under weak data dependence. While the accuracy of covariance matrix estimation corresponding to independent data has been well understood, much less is known in the case of dependent data. We make a step toward filling this gap and establish deviation bounds that depend

Let \((\Omega , \mathcal {F}, (\mathcal {F})_{t\ge 0}, P)\) be a complete stochastic basis, and X be a semimartingale with predictable compensator \((B, C, \nu )\). Consider a family of probability measures \(\mathbf {P}=( {P}^{n, \psi }, \psi \in \Psi , n\ge 1)\), where \(\Psi \) is an index set, \( {P}^{n, \psi }{\mathop {\ll }\limits ^\mathrm{loc}}{P}\), and denote the likelihood ratio process by

We consider the problem of Hurst index estimation for solutions of stochastic differential equations driven by an additive fractional Brownian motion. Using techniques of the Malliavin calculus, we analyze the asymptotic behavior of the quadratic variations of the solution, defined via higher-order increments. Then we apply our results to construct and study estimators for the Hurst index.

We give a new integral characterization of the Dirichlet process on a general phase space. To do so, we first prove a characterization of the nonsymmetric Beta distribution via size-biased sampling. Two applications are a new characterization of the Dirichlet distribution and a marked version of a classical characterization of the Poisson–Dirichlet distribution via invariance and independence properties

We introduce the free analogue of the classical beta prime distribution by the multiplicative free convolution of the free Poisson and the reciprocal of free Poisson distributions, and related free analogues of the classical F, T, and beta distributions. We show the rationales of our free analogues via the score functions and the potentials. We calculate the moments of the free beta prime distribution

We consider two directed polymer models in the Kardar–Parisi–Zhang (KPZ) universality class: the O’Connell–Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m, n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf–Cole solution of the KPZ equation with the corresponding (m, n)-spiked initial condition. This new

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous

The Mallows measure is a probability measure on \(S_n\) where the probability of a permutation \(\pi \) is proportional to \(q^{l(\pi )}\) with \(q > 0\) being a parameter and \(l(\pi )\) the number of inversions in \(\pi \). We show the convergence of the random empirical measure of the product of two independent permutations drawn from the Mallows measure, when q is a function of n and \(n(1-q)\)

In this paper, we study convergence of random walks, on finite quantum groups, arising from linear combination of irreducible characters. We bound the distance to the Haar state and determine the asymptotic behavior, i.e., the limit state if it exists. We note that the possible limits are any central idempotent state. We also look at cutoff phenomenon in the Sekine finite quantum groups.

In this paper, we study the central limit theorem and its functional form for random fields which are started not from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct

We obtain a lower bound for the coarse Ricci curvature of continuous-time pure-jump Markov processes, with an emphasis on interacting particle systems. Applications to several models are provided, with a detailed study of the herd behavior of a simple model of interacting agents.

Using the tools of stochastic analysis, we prove various gradient estimates and Harnack inequalities for Feynman–Kac semigroups with possibly unbounded potentials. One of the main results is a derivative formula which can be used to characterize a lower bound on Ricci curvature using a potential.

We consider in this work a model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states and is thus degenerate. We characterize the long-time limit of our model system as the perturbation parameter tends to zero. The degeneracy in our model system carries features found in some partial differential

A new family of stable processes indexed by metric spaces with stationary increments is introduced. They are special cases of a new family of set-indexed stable processes with Chentsov representation. At the heart of the representation, a result on the so-called measure definite kernels is of independent interest. A limit theorem for set-indexed processes is also established.

This paper concerns a scaling limit of a one-dimensional random walk \(S^x_n\) started from x on the integer lattice conditioned to avoid a non-empty finite set A, the random walk being assumed to be irreducible and have zero mean. Suppose the variance \(\sigma ^2\) of the increment law is finite. Given positive constants b, c and T, we consider the scaled process \(S^{b_N}_{[tN]}/\sigma \sqrt{N}\)

We study the distribution of entries of a random permutation matrix under a “randomized basis,” i.e., we conjugate the random permutation matrix by an independent random orthogonal matrix drawn from Haar measure. It is shown that under certain conditions, the linear combination of entries of a random permutation matrix under a “randomized basis” converges to a sum of independent variables \(sY+Z\)

Free regular convolution semigroups describe the distribution of free subordinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Lévy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions, and we establish an integral identity linking the two semigroups. We provide

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter \( H \in \left( 1/4, 1/2 \right) \). Toward this end, we apply Doss–Sussmann representation of the solution and an approximation of this representation using a first-order Taylor expansion. The obtained rate of convergence is \(n^{-2H +\rho }\), for \(\rho \)

We consider a random symmetric matrix \(\mathbf{X}= [X_{jk}]_{j,k=1}^n\) with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that \( \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4+\delta } < \infty , \delta > 0\), it was proved in Götze et al. (Bernoulli 24(3):2358–2400, 2018) that with high probability the typical distance between the Stieltjes

In this article, we close a gap in the literature by proving existence of invariant measures for reflected stochastic partial differential equations with only one reflecting barrier. This is done by arguing that the sequence \((u(t,\cdot ))_{t \ge 0}\) is tight in the space of probability measures on continuous functions and invoking the Krylov–Bogolyubov theorem. As we no longer have an a priori bound

We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton–Watson trees. We also show the existence of the speed for such rotor walks. More precisely, we show that on the classes of trees under consideration, even in the case when the rotor walk is recurrent, the range grows at linear speed.

Consider the product \(X = X_{1}\cdots X_{m}\) of m independent \(n\times n\) iid random matrices. When m is fixed and the dimension n tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of X for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on m (the number of factor matrices) or on the distribution

Given a target distribution \(\mu \) and a proposal chain with generator Q on a finite state space, in this paper, we study two types of Metropolis–Hastings (MH) generator \(M_1(Q,\mu )\) and \(M_2(Q,\mu )\) in a continuous-time setting. While \(M_1\) is the classical MH generator, we define a new generator \(M_2\) that captures the opposite movement of \(M_1\) and provide a comprehensive suite of

We introduce a new type of random walk where the definition of edge repellence/reinforcement is very different from the one in the “traditional” reinforced random walk models and investigate its basic properties, such as null versus positive recurrence, transience, as well as the speed. The two basic cases will be dubbed “impatient” and “ageing” random walks.