Showing papers in "Journal of Theoretical Probability in 2019"

Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts

Abstract: In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations with Holder–Dini continuous drifts. The key contributions are as follows: (i) by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler–Maruyama scheme for a class of stochastic differential equations which allow the drifts to be Dini continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler–Maruyama scheme for a range of degenerate stochastic differential equations where the drifts are Holder–Dini continuous of order $$\frac{2}{3}$$ with respect to the first component and are merely Dini-continuous concerning the second component.

A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results

Abstract: The consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann–Jorgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvention is to formulate bootstrap consistency in terms of the latter distance between what might be called a conditional law of the (non-measurable) bootstrap process and the law of the limiting process. The main contribution of this note is to provide an equivalent formulation of bootstrap consistency in the space of bounded functions which is more intuitive and easy to work with. Essentially, the equivalent formulation consists of (unconditional) weak convergence of the original process jointly with two bootstrap replicates. As a by-product, we provide two equivalent formulations of bootstrap consistency for statistics taking values in separable metric spaces: the first in terms of (unconditional) weak convergence of the statistic jointly with its bootstrap replicates, the second in terms of convergence in probability of the empirical distribution function of the bootstrap replicates. Finally, the asymptotic validity of bootstrap-based confidence intervals and tests is briefly revisited, with particular emphasis on the (in practice, unavoidable) Monte Carlo approximation of conditional quantiles.

Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

A Smooth Transition from Wishart to GOE

Abstract: It is well known that an $$n \times n$$ Wishart matrix with d degrees of freedom is close to the appropriately centered and scaled Gaussian orthogonal ensemble (GOE) if d is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when $$d = \Theta ( n^{3} )$$ . Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when $$d / n^{3} \rightarrow c \in (0, \infty )$$ . This shows, in particular, that the phase transition from Wishart to GOE is smooth.

The Defect of Random Hyperspherical Harmonics

Abstract: Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-sphere ( $$d\ge 2$$ ). We investigate the distribution of their defect, i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance (Marinucci and Wigman in J Phys A Math Theor 44:355206, 2011) and a central limit theorem (Marinucci and Wigman in Commun Math Phys 327(3):849–872, 2014), both in the high-energy limit. Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener–Ito chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein–Malliavin approximation techniques by Nourdin and Peccati (in Prob Theory Relat Fields 145(1–2):75–118, 2009; Normal approximations with Malliavin calculus. Cambridge Tracts in Mathematics, vol 192, Cambridge University Press, Cambridge, 2012). Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.

Derivatives of Feynman-Kac Semigroups

Abstract: We prove Bismut-type formulae for the first and second derivatives of a Feynman–Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, although the assumptions are purely geometric.

Global Spectrum Fluctuations for Gaussian Beta Ensembles: A Martingale Approach

Abstract: The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $$\beta $$ is allowed to vary with the matrix size n. In particular, we show that as $$n \rightarrow \infty $$ with $$n\beta \rightarrow \infty $$ , the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes

Abstract: The Ornstein–Uhlenbeck process with reflection, which has been the subject of an enormous body of literature, both theoretical and applied, is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. In this work, we are mainly concerned with the study of the asymptotic behavior of the trajectory fitting estimator for nonergodic reflected Ornstein–Uhlenbeck processes, including strong consistency and asymptotic distribution. Moreover, we also prove that this kind of estimator for ergodic reflected Ornstein–Uhlenbeck processes does not possess the property of strong consistency.

Higher-Order Derivative of Intersection Local Time for Two Independent Fractional Brownian Motions

Abstract: In this article, we obtain sharp conditions for the existence of the high-order derivatives (k-th order) of intersection local time $$ \widehat{\alpha }^{(k)}(0)$$ of two independent d-dimensional fractional Brownian motions $$B^{H_1}_t$$ and $$\widetilde{B}^{H_2}_s$$ of Hurst parameters $$H_1$$ and $$H_2$$ , respectively. We also study their exponential integrability.

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

Abstract: Consider a random process s that is a solution of the stochastic differential equation $$\mathrm {L}s = w$$ with $$\mathrm {L}$$ a homogeneous operator and w a multidimensional Levy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on $$\mathrm {L}$$ and w such that $$a^H s(\cdot / a)$$ converges in law to a non-trivial self-similar process for some H, when $$a \rightarrow 0$$ (coarse-scale behavior) or $$a \rightarrow \infty $$ (fine-scale behavior). The parameter H depends on the homogeneity order of the operator $$\mathrm {L}$$ and the Blumenthal–Getoor and Pruitt indices associated with the Levy white noise w. Finally, we apply our general results to several famous classes of random processes and random fields and illustrate our results on simulations of Levy processes.

Mild Solutions and Harnack Inequality for Functional Stochastic Partial Differential Equations with Dini Drift

Abstract: The existence and uniqueness of a mild solution for a class of functional stochastic partial differential equations with multiplicative noise and a locally Dini continuous drift are proved. In addition, under a reasonable condition the solution is non-explosive. Moreover, Harnack inequalities are derived for the associated semigroup under certain global conditions, which is new even in the case without delay.

Viability for Stochastic Differential Equations Driven by G-Brownian Motion

Abstract: In this paper, we prove a type of Nagumo theorem on viability properties for stochastic differential equations driven by G-Brownian motion (G-SDEs). In particular, an equivalent criterion is formulated through stochastic contingent and tangent sets. Moreover, by the approach of direct and inverse images for stochastic tangent sets we present checkable conditions which keep the solution of a given G-SDE evolving in some particular sets.

On Functional Records and Champions

Abstract: A record among a sequence of iid random variables $$X_1,X_2,\dots $$ on the real line is defined as a member $$X_n$$ such that $$X_n>\max (X_1,\cdots ,X_{n-1})$$ . Trying to generalize this concept to random vectors, or even stochastic processes with continuous sample paths, we introduce two different concepts: A simple record is a stochastic process (or a random vector) $${\varvec{X}}_n$$ that is larger than $${\varvec{X}}_1,\cdots ,{\varvec{X}}_{n-1}$$ in at least one component, whereas a complete record has to be larger than its predecessors in all components. In particular, the probability that a stochastic process $${\varvec{X}}_n$$ is a record as n tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the conditional distribution of $${\varvec{X}}_n$$ given that $${\varvec{X}}_n$$ is a record is derived.

Centers of probability measures without the mean

Abstract: In the recent years, the notion of mixability has been developed with applications to operations research, optimal transportation, and quantitative finance. An n-tuple of distributions is said to be jointly mixable if there exist n random variables following these distributions and adding up to a constant, called center, with probability one. When the n distributions are identical, we speak of complete mixability. If each distribution has finite mean, the center is obviously the sum of the means. In this paper, we investigate the set of centers of completely and jointly mixable distributions not having a finite mean. In addition to several results, we show the (possibly counterintuitive) fact that, for each $$n \ge 2$$ , there exist n standard Cauchy random variables adding up to a constant C if and only if $$\begin{aligned} |C|\le \frac{n\,\log (n-1)}{\pi }. \end{aligned}$$

Empirical Distributions of Eigenvalues of Product Ensembles

Abstract: Assume a finite set of complex random variables form a determinantal point process; we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to two types of n-by-n random matrices as n goes to infinity. The first one is the product of m i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of m independent Haar unitary matrices with sizes $$n_j\times n_j$$ for $$1\le j \le m$$ . Assuming m depends on n, by using the special structures of the eigenvalues we developed, explicit limits of spectral distributions are obtained regardless of the speed of m compared to n. For the product of m Ginibre ensembles, as m is fixed, the limiting distribution is known by various authors, e.g., Gotze and Tikhomirov (On the asymptotic spectrum of products of independent random matrices, 2010. http://arxiv.org/pdf/1012.2710v3.pdf ), Bordenave (Electron Commun Probab 16:104–113, 2011), O’Rourke and Soshnikov (Electron J Probab 16(81):2219–2245, 2011) and O’Rourke et al. (J Stat Phys 160(1):89–119, 2015). Our results hold for any $$m\ge 1$$ which may depend on n. For the product of truncations of Haar-invariant unitary matrices, we show a rich feature of the limiting distribution as $$n_j/n$$ ’s vary. In addition, some general results on arbitrary rotation-invariant determinantal point processes are also derived. In particular, we obtain an inequality for the fourth moment of linear statistics of complex random variables forming a determinantal point process. This inequality is known for the complex Ginibre ensemble only (Hwang in Random matrices and their applications (Brunswick, Maine, 1984), Contemporary Mathematics, American Mathematics Society, Providence, vol 50, pp 145–152, 1986). Our method is the determinantal point process rather than the contour integral by Hwang.

Realizations and Factorizations of Positive Definite Kernels

Abstract: Given a fixed sigma-finite measure space $$\left( X,\mathscr {B}, u \right) $$ , we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $$ u $$ is infinite, but sigma-finite. We introduce such positive definite kernels $$K\left( \cdot ,\cdot \right) $$ with the two variables from the subclass of the sigma-algebra $$\mathscr {B}$$ whose elements are sets with finite $$ u $$ measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in $$L^{2}\left( u \right) $$ . Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.

Strong Laws of Large Numbers for Intermediately Trimmed Sums of i.i.d. Random Variables with Infinite Mean

Abstract: We show that for every sequence of nonnegative i.i.d. random variables with infinite mean there exists a proper moderate trimming such that for the trimmed sum process a non-trivial strong law of large numbers holds. We provide an explicit procedure to find a moderate trimming sequence even if the underlying distribution function has a complicated structure, e.g., has no regularly varying tail distribution.

Large deviations for cascades of diffusions arising in oscillating systems of interacting Hawkes processes

Abstract: We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Locherbach (Stoch Process Appl 2017, http://arxiv.org/abs/1512.00265 ) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high-dimensional chain of differential equations driven by 2-dimensional Brownian motion. We study the large population, i.e., small noise limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.

Low-Degree Factors of Random Polynomials

Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocalization of the roots of a random polynomial can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers. We apply our main result to a number of models of random polynomials, including characteristic polynomials of random matrices, where strong delocalization results are known.

On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations

Abstract: 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 comparison results ranging from hitting time and mixing time to asymptotic variance, large deviations and capacity, which demonstrate that $$M_2$$ enjoys superior mixing properties than $$M_1$$. To see that $$M_1$$ and $$M_2$$ are natural transformations, we offer an interesting geometric interpretation of $$M_1$$, $$M_2$$ and their convex combinations as $$\ell ^1$$ minimizers between Q and the set of $$\mu $$-reversible generators, extending the results by Billera and Diaconis (Stat Sci 16(4):335–339, 2001). We provide two examples as illustrations. In the first one, we give explicit spectral analysis of $$M_1$$ and $$M_2$$ for Metropolized independent sampling, while in the second example, we prove a Laplace transform order of the fastest strong stationary time between birth–death $$M_1$$ and $$M_2$$.

Empirical Spectral Distribution of a Matrix Under Perturbation

Abstract: We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.

Existence and Uniqueness of Quasi-stationary Distributions for Symmetric Markov Processes with Tightness Property

Abstract: Let X be an irreducible symmetric Markov process with the strong Feller property. We assume, in addition, that X is explosive and has a tightness property. We then prove the existence and uniqueness of quasi-stationary distributions of X.

Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment

Abstract: We consider branching processes in discrete time for structured population in varying environment. Each individual has a trait which belongs to some general state space and both its reproduction law and the trait inherited by its offsprings may depend on its trait and the environment. We study the long-time behavior of the population and the ancestral lineage of typical individuals under general assumptions. We focus on the mean growth rate and the trait distribution among the population. The approach relies on many-to-one formulae and the analysis of the genealogy, and a key role is played by well-chosen (possibly non-homogeneous) Markov chains. The applications use large deviations principles and the Harris ergodicity for these auxiliary Markov chains.

Lumpings of Algebraic Markov Chains Arise from Subquotients

Abstract: A function on the state space of a Markov chain is a “lumping” if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a “top-to-random-with-standardisation” chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto–Reutenauer algebra.

Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion

Abstract: In this paper, we discuss a class of neutral retarded stochastic functional differential equations driven by a fractional Brownian motion on Hilbert spaces. We develop a $$C_0$$ -semigroup theory of the driving deterministic neutral system and formulate the neutral time delay equation under consideration as an infinite-dimensional stochastic system without time lag and neutral item. Consequently, a criterion is presented to identify a strictly stationary solution for the systems considered. In particular, the ergodicity of the strictly stationary solution is studied. Subsequently, the ergodicity behavior of non-stationary solution for the systems considered is also investigated. We present an example which can be explicitly determined to illustrate our theory in the work.

Improved mixing rates of directed cycles by added connection

Abstract: We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced. As a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square-factor improvement of the mixing rate compared to the reversible version of the Markov chain.

Third Cumulant Stein Approximation for Poisson Stochastic Integrals

Abstract: We derive Edgeworth-type expansions for Poisson stochastic integrals, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain Stein approximation bounds for stochastic integrals, which are based on third cumulants instead of the $$L^3$$ -norm term found in the literature. The use of the third cumulant results in a convergence rate faster than the classical Berry–Esseen rate for certain examples.

Cylindrical Martingale Problems Associated with Lévy Generators

Abstract: We introduce and discuss Levy-type cylindrical martingale problems on separable reflexive Banach spaces. Our main observations are the following: Cylindrical martingale problems have a one-to-one relation to weak solutions of stochastic partial differential equations, and well-posed problems possess the strong Markov property and a Cameron–Martin–Girsanov-type formula holds. As applications, we derive existence and uniqueness results.

Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations

Abstract: We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Ito equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Ito equations solve approximated Stratonovich equations with a certain correction term in the drift.

Pathwise Uniqueness of Non-uniformly Elliptic SDEs with Rough Coefficients

Abstract: In this paper, we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.