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.

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 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 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 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.

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

The topic of this paper is the asymptotic distribution of the entries of random orthogonal matrices distributed according to Haar measure. We examine the total variation distance between the joint distribution of the entries of \(W_n\), the \(p_n \times q_n\) upper-left block of a Haar-distributed matrix, and that of \(p_nq_n\) independent standard Gaussian random variables and show that the total

For a spectrally positive and strictly stable process with index in (1, 2), a series representation is obtained for the joint distribution of the “first passage triple” that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including (1) the joint law of the first passage triple and the pre-passage running supremum

In this article, we introduce an infinite-dimensional analogue of the \(\alpha \)-stable Lévy motion, defined as a Lévy process \(Z=\{Z(t)\}_{t \ge 0}\) with values in the space \({\mathbb {D}}\) of càdlàg functions on [0, 1], equipped with Skorokhod’s \(J_1\) topology. For each \(t \ge 0\), Z(t) is an \(\alpha \)-stable process with sample paths in \({\mathbb {D}}\), denoted by \(\{Z(t,s)\}_{s\in

We define an inhomogeneous percolation model on “ladder graphs” obtained as direct products of an arbitrary graph \(G = (V,E)\) and the set of integers \({\mathbb {Z}}\). (Vertices are thought of as having a “vertical” component indexed by an integer.) We make two natural choices for the set of edges, producing an unoriented graph \({\mathbb {G}}\) and an oriented graph \(\vec {{\mathbb {G}}}\). These

For arbitrary Borel probability measures on the real line, necessary and sufficient conditions are presented that characterize best purely atomic approximations relative to the classical Lévy probability metric, given any number of atoms, and allowing for additional constraints regarding locations or weights of atoms. The precise asymptotics (as the number of atoms goes to infinity) of the approximation

The law of an appropriately scaled sum of p-adic-valued, independent, identically and rotation-symmetrically distributed random variables weakly converges to a semi-stable law, if the tail probabilities of the variables satisfy some assumption. If we consider a scaled sum of such random variables with a sufficiently much higher scaling order, it accumulates to the origin, and the mass of any set not

We obtain almost sure limit theorems for partial maxima of norms of a sequence of Banach-valued Gaussian random variables.

In this paper we study the drifted Brownian meander that is a Brownian motion starting from u and subject to the condition that \( \min _{ 0\le z \le t} B(z)> v \) with \( u > v \). The limiting process for \( u \downarrow v \) is analysed, and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage

According to the Wiener–Hopf factorization, the characteristic function \(\varphi \) of any probability distribution \(\mu \) on \(\mathbb {R}\) can be decomposed in a unique way as $$\begin{aligned} 1-s\varphi (t)=[1-\chi _-(s,it)][1-\chi _+(s,it)],\quad |s|\le 1,\,t\in \mathbb {R}\,, \end{aligned}$$ where \(\chi _-(e^{iu},it)\) and \(\chi _+(e^{iu},it)\) are the characteristic functions of possibly