Abstract: Abstract The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to \(2/(\pi N).\) As a consequence, we prove strong laws and a limit distribution for the local time. PubDate: 2021-01-19

Abstract: Abstract The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer–Major theorem, asymptotic behavior of the third and fourth cumulants of fBm and the optimal fourth moment theorem. PubDate: 2021-01-18

Abstract: Abstract In this paper, we introduce a specific kind of doubly reflected backward stochastic differential equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki’s condition, we show the existence of the solution (in consideration of the driver’s nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart we provide a suitable a priori estimates which immediately implies the uniqueness of the solution. PubDate: 2021-01-13

Abstract: Abstract The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power-law shape function and non-stationary noises with a power-law variance function. In this paper, we study sample path properties of the generalized fractional Brownian motion, including Hölder continuity, path differentiability/non-differentiability, and functional and local law of the iterated logarithms. PubDate: 2021-01-09

Abstract: Abstract Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin–Wentzell small noise asymptotics. Different levels of intensity of the small common noise lead to different types of large deviation behavior, and we provide a precise characterization of the various regimes. The rate functions can be interpreted as the value functions of certain stochastic control problems in which there are two types of controls; one of the controls is random and nonanticipative and arises from the aggregated contributions of the individual Brownian noises, whereas the second control is nonrandom and corresponds to the small common Brownian noise that impacts all particles. We also study large deviation behavior of interacting particle systems approximating various types of Feynman–Kac functionals. Proofs are based on stochastic control representations for exponential functionals of Brownian motions and on uniqueness results for weak solutions of stochastic differential equations associated with controlled nonlinear Markov processes PubDate: 2021-01-09

Abstract: Abstract This paper concerns the density of the Hartman–Watson law. Yor (Z Wahrsch Verw Gebiete 53:71–95, 1980) obtained an integral formula that gives a closed-form expression of the Hartman–Watson density. In this paper, based on Yor’s formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a result of Dufresne (Adv Appl Probab 33:223–241, 2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion. PubDate: 2021-01-07

Abstract: Abstract Under the scenario of high-frequency data, a consistent estimator of the realized Laplace transform of volatility is proposed by Todorov and Tauchen (Econometrica 80:1105–1127, 2012) and a related central limit theorem has been well established. In this paper, we investigate the asymptotic tail behaviour of the empirical realized Laplace transform of volatility (ERLTV). We establish both a large deviation principle and a moderate deviation principle for the ERLTV. The good rate function for the large deviation principle is well defined in the whole real space, which indicates a limit for the normalized logarithmic tail probability of the ERLTV. Moreover, we also derive the function-level large and moderate deviation principles for ERLTV. PubDate: 2021-01-06

Abstract: Abstract We study a class of stationary determinantal processes on configurations of N labeled objects that may be present or absent at each site of \({\mathbb {Z}}^d\) . Our processes, which include the uniform spanning forest as a principal example, arise from the block Toeplitz matrices of matrix-valued functions on the d-torus. We find the maximum level of uniform insertion tolerance for these processes, extending a result of Lyons and Steif from the \(N = 1\) case to \(N > 1\) . We develop a method for conditioning determinantal processes in the general discrete setting to be as large as possible in a fixed set as an approach to determining uniform insertion tolerance. The method of conditioning on maximality developed here is used in a subsequent paper to study stochastic domination, strong domination and phase uniqueness for the same class of processes. PubDate: 2021-01-02

Abstract: Abstract We investigate a Belinschi–Nica-type semigroup for free and Boolean max-convolutions. We prove that this semigroup at time one connects limit theorems for freely and Boolean max-infinitely divisible distributions. Moreover, we also construct a max-analogue of Boolean-classical Bercovici–Pata bijection, establishing the equivalence of limit theorems for Boolean and classical max-infinitely divisible distributions. PubDate: 2021-01-02

Abstract: Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution \(\mu \) , the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of \(\mu \) is obtained by a function of independent random variables. Two open problems are also asked. PubDate: 2021-01-01

Abstract: Abstract The paper deals with some properties of set-valued functions having bounded Riesz p-variation. Set-valued integrals of Young type for such multifunctions are introduced. Selection results and properties of such set-valued integrals are discussed. These integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion. PubDate: 2020-12-08

Abstract: The duality theory of the Monge–Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let \((\mathcal {X},\mathcal {F},\mu )\) and \((\mathcal {Y},\mathcal {G},\nu )\) be any probability spaces and \(c:\mathcal {X}\times \mathcal {Y}\rightarrow \mathbb {R}\) a measurable cost function such that \(f_1+g_1\le c\le f_2+g_2\) for some \(f_1,\,f_2\in L_1(\mu )\) and \(g_1,\,g_2\in L_1(\nu )\) . Define \(\alpha (c)=\inf _P\int c\,dP\) and \(\alpha ^*(c)=\sup _P\int c\,dP\) , where \(\inf \) and \(\sup \) are over the probabilities P on \(\mathcal {F}\otimes \mathcal {G}\) with marginals \(\mu \) and \(\nu \) . Some duality theorems for \(\alpha (c)\) and \(\alpha ^*(c)\) , not requiring \(\mu \) or \(\nu \) to be perfect, are proved. As an example, suppose \(\mathcal {X}\) and \(\mathcal {Y}\) are metric spaces and \(\mu \) is separable. Then, duality holds for \(\alpha (c)\) (for \(\alpha ^*(c)\) ) provided c is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both \(\alpha (c)\) and \(\alpha ^*(c)\) if the maps \(x\mapsto c(x,y)\) and \(y\mapsto c(x,y)\) are continuous, or if c is bounded and \(x\mapsto c(x,y)\) is continuous. This improves the existing results in Ramachandran and Ruschendorf (Probab Theory Relat Fields 101:311–319, 1995) if c satisfies the quoted conditions and the cardinalities of \(\mathcal {X}\) and \(\mathcal {Y}\) do not exceed the continuum. PubDate: 2020-12-01

Abstract: Abstract For an arbitrary transient random walk \((S_n)_{n\ge 0}\) in \({\mathbb {Z}}^d\) , \(d\ge 1\) , we prove a strong law of large numbers for the spatial sum \(\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))\) of a function f of the local times \(l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}\) . Particular cases are the number of visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function \(f(i)={\mathbb {I}}\{i\ge 1\}\) ; \(\alpha \) -fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to \(f(i)=i^\alpha \) ; sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where \(f(i)={\mathbb {I}}\{i=j\}\) . PubDate: 2020-12-01

Abstract: Abstract We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (C R Acad Sci Paris Sér I Math 300:281–282, 1985) for Brownian motion and that of Song et al. (Electron Commun Probab 23:10, 2018) for \(\alpha \) -stable Lévy processes with \(1<\alpha <2\) . Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes. PubDate: 2020-12-01

Abstract: Abstract In this article, we investigate the percolative properties of Brownian interlacements, a model introduced by Sznitman (Bull Braz Math Soc New Ser 44(4):555–592, 2013), and show that: the interlacement set is “well-connected”, i.e., any two “sausages” in d-dimensional Brownian interlacements, \(d\ge 3\) , can be connected via no more than \(\lceil (d-4)/2 \rceil \) intermediate sausages almost surely; while the vacant set undergoes a non-trivial percolation phase transition when the level parameter varies. PubDate: 2020-12-01

Abstract: Abstract We study the extremes for a class of a symmetric stable random fields with long-range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of càdlàg functions of several variables. The limits in both types of theorems are of a new kind, and only in a certain range of parameters, these limits have the Fréchet distribution. PubDate: 2020-12-01

Abstract: Abstract The \(\text {PD}_\alpha ^{(r)}\) distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the \(\text {PD}_\alpha \) distribution introduced by Kingman, to which it reduces when \(r=0\) . The parameter \(\alpha \in (0,1)\) arises from its construction based on ratios of ordered jumps of an \(\alpha \) -stable subordinator, and the parameter \(r>0\) signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution \(\text {PD}(\theta )\) , occur as limiting cases of \(\text {PD}_\alpha ^{(r)}\) , as \(r\rightarrow \infty \) . As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made. PubDate: 2020-12-01

Abstract: Abstract We consider a time-inhomogeneous Markov process \(X = (X_t)_t\) with jumps having state-dependent jump intensity, with values in \({\mathbb {R}}^d , \) and we are interested in its longtime behavior. The infinitesimal generator of the process is given for any sufficiently smooth test function f by $$\begin{aligned} L_t f (x) = \sum _{i=1}^d \frac{\partial f}{\partial x_i } (x) b^i ( t,x) + \int _{{\mathbb {R}}^m } [ f ( x + c ( t, z, x)) - f(x)] \gamma ( t, z, x) \mu (\mathrm{d}z ) , \end{aligned}$$ where \( \mu \) is a \(\sigma \) -finite measure on \(({\mathbb {R}}^m , {\mathcal B} ( {\mathbb {R}}^m ) ) \) describing the jumps of the process. We give conditions on the coefficients b(t, x) , c(t, z, x) and \( \gamma ( t, z, x ) \) under which the longtime behavior of X can be related to the longtime behavior of a time-homogeneous limit process \({\bar{X}} . \) Moreover, we introduce a coupling method for the limit process which is entirely based on certain of its big jumps and which relies on the regeneration method. We state explicit conditions in terms of the coefficients of the process allowing control of the speed of convergence to equilibrium both for X and for \({\bar{X}}\) . PubDate: 2020-12-01

Abstract: Abstract Regular variation is an essential condition for the existence of a Darling–Kac law. We weaken this condition assuming that the renewal distribution belongs to the domain of geometric partial attraction of a semistable law. In the simple setting of one-sided null recurrent renewal chains, we derive a Darling–Kac limit theorem along subsequences. Also in this context, we determine the asymptotic behaviour of the renewal function and obtain a Karamata theorem for positive operators. We provide several examples of dynamical systems to which these results apply. PubDate: 2020-12-01

Abstract: Abstract The so-called supOU processes, namely the superpositions of Ornstein–Uhlenbeck type processes, are stationary processes for which one can specify separately the marginal distribution and the temporal dependence structure. They can have finite or infinite variance. We study the limit behavior of integrated infinite variance supOU processes adequately normalized. Depending on the specific circumstances, the limit can be fractional Brownian motion but it can also be a process with infinite variance, a Lévy stable process with independent increments or a stable process with dependent increments. We show that it is even possible to have infinite variance integrated supOU processes converging to processes whose moments are all finite. A number of examples are provided. PubDate: 2020-12-01