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On the Local Time of the Half-Plane Half-Comb Walk J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-19 Endre Csáki, Antónia Földes
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
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How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion? J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-18 Ehsan Azmoodeh, Yuliya Mishura, Farzad Sabzikar
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
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Doubly Reflected Backward Stochastic Differential Equations in the Predictable Setting J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-13 Ihsan Arharas, Siham Bouhadou, Youssef Ouknine
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
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Path Properties of a Generalized Fractional Brownian Motion J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-09 Tomoyuki Ichiba, Guodong Pang, Murad S. Taqqu
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
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Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-09 Amarjit Budhiraja, Michael Conroy
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
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Integral Representations for the Hartman–Watson Density J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-07 Yuu Hariya
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,
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Large Deviation Principles of Realized Laplace Transform of Volatility J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-06 Xinwei Feng, Lidan He, Zhi Liu
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
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Stationary Determinantal Processes on $${\mathbb {Z}}^d$$ Z d with N Labeled Objects per Site, Part I: Basic Properties and Full Domination J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-02 Justin Cyr
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
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Limit Theorems for Classical, Freely and Boolean Max-Infinitely Divisible Distributions J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-02 Yuki Ueda
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
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Statistical Properties of Eigenvalues of Laplace–Beltrami Operators J. Theor. Probab. (IF 0.682) Pub Date : 2021-01-01 Tiefeng Jiang, Ke Wang
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
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Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals J. Theor. Probab. (IF 0.682) Pub Date : 2020-12-08 Mariusz Michta, Jerzy Motyl
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
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On Asymptotic Properties of Bell Polynomials and Concentration of Vertex Degree of Large Random Graphs J. Theor. Probab. (IF 0.682) Pub Date : 2020-10-04 O. Khorunzhiy
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
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Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-24 S. Franceschi
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
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Compound Poisson Approximations in $$\ell _p$$ ℓ p -norm for Sums of Weakly Dependent Vectors J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-20 V. Čekanavičius, P. Vellaisamy
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.
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Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-14 Shiyu Song
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
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A Central Limit Theorem for Star-Generators of $${S}_{\infty }$$ S ∞ , Which Relates to the Law of a GUE Matrix J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-14 Claus Köstler, Alexandru Nica
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
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Euler–Maruyama Approximations for Stochastic McKean–Vlasov Equations with Non-Lipschitz Coefficients J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-13 Xiaojie Ding, Huijie Qiao
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.
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Backward Stochastic Differential Equations Driven by G -Brownian Motion with Double Reflections J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-13 Hanwu Li, Yongsheng Song
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
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Large Deviations of the Range of the Planar Random Walk on the Scale of the Mean J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-11 Jingjia Liu, Quirin Vogel
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.
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Quaternionic Brownian Windings J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-08 Fabrice Baudoin, Nizar Demni, Jing Wang
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
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Limiting Spectral Radii of Circular Unitary Matrices Under Light Truncation J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-05 Yu Miao, Yongcheng Qi
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
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Pathwise Uniqueness and Non-explosion Property of Skorohod SDEs with a Class of Non-Lipschitz Coefficients and Non-smooth Domains J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-04 Masanori Hino, Kouhei Matsuura, Misaki Yonezawa
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.
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Martingale Nature and Laws of the Iterated Logarithm for Markov Processes of Pure-Jump Type J. Theor. Probab. (IF 0.682) Pub Date : 2020-09-03 Yuichi Shiozawa, Jian Wang
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.
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1-Meixner Random Vectors J. Theor. Probab. (IF 0.682) Pub Date : 2020-07-20 Aurel I. Stan, Florin Catrina
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
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Self-Standardized Central Limit Theorems for Trimmed Lévy Processes J. Theor. Probab. (IF 0.682) Pub Date : 2020-07-14 David M. Mason
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
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Regenerativity of Viterbi Process for Pairwise Markov Models J. Theor. Probab. (IF 0.682) Pub Date : 2020-07-14 Jüri Lember, Joonas Sova
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
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Hörmander’s Hypoelliptic Theorem for Nonlocal Operators J. Theor. Probab. (IF 0.682) Pub Date : 2020-07-13 Zimo Hao, Xuhui Peng, Xicheng Zhang
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
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On the Convergence of Series of Dependent Random Variables J. Theor. Probab. (IF 0.682) Pub Date : 2020-06-27 Safari Mukeru
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
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A General Version of Price’s Theorem J. Theor. Probab. (IF 0.682) Pub Date : 2020-06-26 Felix Voigtlaender
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
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Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities J. Theor. Probab. (IF 0.682) Pub Date : 2020-06-12 Friedrich Götze, Holger Sambale, Arthur Sinulis
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
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Maximum Drawdown and Drawdown Duration of Spectrally Negative Lévy Processes Decomposed at Extremes J. Theor. Probab. (IF 0.682) Pub Date : 2020-06-06 Ceren Vardar-Acar, Mine Çağlar, Florin Avram
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
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Correction to: Cylindrical Martingale Problems Associated with Lévy Generators J. Theor. Probab. (IF 0.682) Pub Date : 2020-06-03 David Criens
In this note, we correct claims made in.
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On the Long-Range Dependence of Mixed Fractional Poisson Process J. Theor. Probab. (IF 0.682) Pub Date : 2020-06-02 K. K. Kataria, M. Khandakar
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.
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A Penalised Model Reproducing the Mod-Poisson Fluctuations in the Sathé–Selberg Theorem J. Theor. Probab. (IF 0.682) Pub Date : 2020-03-27 Yacine Barhoumi-Andréani
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
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Self-similar Growth Fragmentations as Scaling Limits of Markov Branching Processes J. Theor. Probab. (IF 0.682) Pub Date : 2019-12-17 Benjamin Dadoun
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.
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Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space J. Theor. Probab. (IF 0.682) Pub Date : 2019-12-05 C. A. Fonseca-Mora
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
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Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths J. Theor. Probab. (IF 0.682) Pub Date : 2019-11-25 Benjamin Gess; Cheng Ouyang; Samy Tindel
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
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Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach J. Theor. Probab. (IF 0.682) Pub Date : 2019-11-07 Alessandra Cipriani, Jan de Graaff, Wioletta M. Ruszel
In this paper we investigate scaling limits of the odometer in divisible sandpiles on d-dimensional tori following up the works of Chiarini et al. (Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits, 2018), Cipriani et al. (Probab Theory Relat Fields 172:829–868, 2017; Stoch Process Appl 128(9):3054–3081, 2018). Relaxing the assumption of independence of the weights of
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Talagrand Inequality at Second Order and Application to Boolean Analysis J. Theor. Probab. (IF 0.682) Pub Date : 2019-11-04 Kevin Tanguy
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.
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Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data J. Theor. Probab. (IF 0.682) Pub Date : 2019-11-02 Tiefeng Jiang, Junshan Xie
Let \({X}_{k}=(x_{k1}, \ldots , x_{kp})', k=1,\ldots ,n\), be a random sample of size n coming from a p-dimensional population. For a fixed integer \(m\ge 2\), consider a hypercubic random tensor \(\mathbf {{T}}\) of mth order and rank n with $$\begin{aligned} \mathbf {{T}}= \sum _{k=1}^{n}\underbrace{{X}_{k}\otimes \cdots \otimes {X}_{k}}_{\mathrm{multiplicity}\ m}=\Big (\sum _{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots
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Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-31 Hyunchul Park, Yimin Xiao, Xiaochuan Yang
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
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Bridges with Random Length: Gamma Case J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-26 Mohamed Erraoui; Astrid Hilbert; Mohammed Louriki
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
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Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-25 Zhongquan Tan,Shengchao Zheng
Let \(\{Z(\tau ,s), (\tau ,s)\in [a,b]\times [0,T]\}\) with some positive constants a , b , T be a centered Gaussian random field with variance function \(\sigma ^{2}(\tau ,s)\) satisfying \(\sigma ^{2}(\tau ,s)=\sigma ^{2}(\tau )\). We first derive the exact tail asymptotics (as \(u \rightarrow \infty \)) for the probability that the maximum \(M_H(T) = \max _{(\tau , s) \in [a, b] \times [0, T]}
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Extreme Value Theory for Long-Range-Dependent Stable Random Fields J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-25 Zaoli Chen, Gennady Samorodnitsky
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.
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Pinned Diffusions and Markov Bridges J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-23 Florian Hildebrandt; Sylvie Rœlly
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
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Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-17 David Criens
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
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On the Convergence of FK–Ising Percolation to $$\mathrm {SLE}(16/3, (16/3)-6)$$SLE(16/3,(16/3)-6) J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-10 Christophe Garban; Hao Wu
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
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Total Variation Cutoff for the Transpose Top-2 with Random Shuffle J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-05 Subhajit Ghosh
In this paper, we investigate the properties of a random walk on the alternating group \(A_n\) generated by three cycles of the form \((i,n-1,n)\) and \((i,n,n-1)\). We call this the transpose top-2 with random shuffle. We find the spectrum of the transition matrix of this shuffle. We show that the mixing time is of order \(\left( n-\frac{3}{2}\right) \log n\) and prove that there is a total variation
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Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-05 Xiequan Fan; Ion Grama; Quansheng Liu
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,
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A Spectral Characterization for Concentration of the Cover Time J. Theor. Probab. (IF 0.682) Pub Date : 2019-10-01 Jonathan Hermon
We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average
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Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-30 E. Löcherbach
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
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On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-27 Na Zhang, Lucas Reding, Magda Peligrad
Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (J Theor Probab, 2018. arXiv:1802.09106) showed that the central limit theorem (CLT) holds for stationary ortho-martingale random fields when they are started from a fixed past trajectory. In this paper, we study this type of behavior, also known under the name of quenched CLT, for a class of random
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Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-24 Anyue Chen
The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes
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Percolative Properties of Brownian Interlacements and Its Vacant Set J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-21 Xinyi Li
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;
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Spectral Radii of Products of Random Rectangular Matrices J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-19 Yongcheng Qi, Mengzi Xie
We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when
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Rate of Convergence for the Weighted Hermite Variations of the Fractional Brownian Motion J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-10 Nicholas Ma, David Nualart
In this paper, we obtain a rate of convergence in the central limit theorem for high order weighted Hermite variations of the fractional Brownian motion. The proof is based on the techniques of Malliavin calculus and the quantitative stable limit theorems proved by Nourdin et al. (Ann Probab 44:1–41, 2016).
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Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles J. Theor. Probab. (IF 0.682) Pub Date : 2019-09-10 Deng Zhang
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
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Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2 J. Theor. Probab. (IF 0.682) Pub Date : 2019-08-29 Endre Csáki, Antónia Földes
We study the path behavior of the simple symmetric walk on some comb-type subsets of \({{\mathbb {Z}}}^2\) which are obtained from \({{\mathbb {Z}}}^2\) by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation results and discuss their consequences.
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Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$ Z d J. Theor. Probab. (IF 0.682) Pub Date : 2019-08-28 Inna M. Asymont, Dmitry Korshunov
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 (a) visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley
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The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case J. Theor. Probab. (IF 0.682) Pub Date : 2019-08-22 Danijel Grahovac, Nikolai N. Leonenko, Murad S. Taqqu
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
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