Poisson Space Research Articles

Normal Approximation of Kabanov–Skorohod Integrals on Poisson Spaces

AbstractWe consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process.

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

AbstractProofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of “stopping set”, not requiring any topological property for the underlying measurable space, as well as on the new concept of a “continuous‐time decision tree”, for which we establish several fundamental properties. We apply our findings to the ‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite ‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is for the planar confetti percolation model with bounded, ‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases.

Algebraic Symplectic Reduction and Quantization of Singular Spaces

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K=Un,SOn. Extending these results to groups of other types is one of the goals of this paper.Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie(K)⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G=KC.We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K, we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K-spaces.An essential tool in our study is the dual Poisson-Lie group K⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K⁎, and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie(K)⁎→K⁎.

A modified Φ-Sobolev inequality for canonical Lévy processes and its applications

A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.

Analysis of space-dependent noise functionals with an application to linearly correlated processes

Recently, Hida et al.5, 11 introduced a noise of new type, called the Poisson space noise. It depends only on the space parameter, quite different from the time derivative of a Poisson process. In this paper, without employing the Minlos theorem, we shall construct Poisson space noise from the view of its path behavior, which will be realized by the space derivative of an additive process. Then the existence of the associated Poisson space noise measure and its measurable support are studied, and an orthogonal decomposition of square-integrable Poisson space noise functionals is achieved. Accordingly, we define the Segal–Bargmann transform of Poisson space noise functionals. Through the Segal–Bargmann transform, the test–generalized function setting is established. Then the Poisson space noise will be thought of as generalized Poisson space noise functionals. Finally, as an application, we shall study a linearly correlated process driven by Poisson space noises with a kernel function of Volterra type.

Quantitative two-scale stabilization on the Poisson space

We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one—that we evaluate and compare at two different scales—and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization—see Penrose and Yukich (Ann. Appl. Probab. 11 (2001) 1005–1041) and Penrose (Ann. Probab. 33 (2005) 1945–1991). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (Ann. Appl. Probab. 27 (2017) 1588–1645) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbour graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.

Multi-dimensional normal approximation of heavy-tailed moving averages

In this paper we extend the refined second-order Poincaré inequality for Poisson functionals from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin–Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavy-tailed moving averages. The extension allows a functional with multivariate arguments, i.e. multiple moving averages and also multivariate values of the functional. Such a set-up has previously not been explored in the framework of stable moving average processes. It can potentially capture probabilistic properties which cannot be described solely by the one-dimensional marginals, but instead require the joint distribution.

Spatio-temporal distribution characteristics of COVID-19 in China: a city-level modeling study

BackgroundThe coronavirus disease 2019 (COVID-19) has become a pandemic. Few studies have been conducted to investigate the spatio-temporal distribution of COVID-19 on nationwide city-level in China.ObjectiveTo analyze and visualize the spatiotemporal distribution characteristics and clustering pattern of COVID-19 cases from 362 cities of 31 provinces, municipalities and autonomous regions in mainland China.MethodsA spatiotemporal statistical analysis of COVID-19 cases was carried out by collecting the confirmed COVID-19 cases in mainland China from January 10, 2020 to October 5, 2020. Methods including statistical charts, hotspot analysis, spatial autocorrelation, and Poisson space–time scan statistic were conducted.ResultsThe high incidence stage of China’s COVID-19 epidemic was from January 17 to February 9, 2020 with daily increase rate greater than 7.5%. The hot spot analysis suggested that the cities including Wuhan, Huangshi, Ezhou, Xiaogan, Jingzhou, Huanggang, Xianning, and Xiantao, were the hot spots with statistical significance. Spatial autocorrelation analysis indicated a moderately correlated pattern of spatial clustering of COVID-19 cases across China in the early phase, with Moran’s I statistic reaching maximum value on January 31, at 0.235 (Z = 12.344, P = 0.001), but the spatial correlation gradually decreased later and showed a discrete trend to a random distribution. Considering both space and time, 19 statistically significant clusters were identified. 63.16% of the clusters occurred from January to February. Larger clusters were located in central and southern China. The most likely cluster (RR = 845.01, P < 0.01) included 6 cities in Hubei province with Wuhan as the centre. Overall, the clusters with larger coverage were in the early stage of the epidemic, while it changed to only gather in a specific city in the later period. The pattern and scope of clusters changed and reduced over time in China.ConclusionsSpatio-temporal cluster detection plays a vital role in the exploration of epidemic evolution and early warning of disease outbreaks and recurrences. This study can provide scientific reference for the allocation of medical resources and monitoring potential rebound of the COVID-19 epidemic in China.

Concentration on Poisson spaces via modified [formula omitted]-Sobolev inequalities

Concentration properties of functionals of general Poisson processes are studied. Using a modified Φ-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentration inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well.

Multidimensional parameter estimation of heavy‐tailed moving averages

AbstractIn this article we present a parametric estimation method for certain multiparameter heavy‐tailed Lévy‐driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one‐dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein–Uhlenbeck process, certain CARMA(2, 1) models, and Ornstein–Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

Almost sure central limit theorems in stochastic geometry

AbstractWe prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals arising in stochastic geometry. As a consequence, we provide almost sure central limit theorems for (i) the total edge length of thek-nearest neighbors random graph, (ii) the clique count in random geometric graphs, and (iii) the volume of the set approximation via the Poisson–Voronoi tessellation.

Stable limit theorems on the Poisson space

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Wasserstein transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by G. Peccati, J. L. Sole, M. S. Taqqu & F. Utzet. “Stein’s method and normal approximation of Poisson functionals” for Gaussian approximations; and by G. Peccati “The Chen-Stein method for Poisson functionals” for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of C. Dobler & G.Peccati “The fourth moment theorem on the Poisson space”. We give an application to stochastic processes.

A Berry–Esseén theorem for partial sums of functionals of heavy-tailed moving averages

In this paper we obtain Berry–Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein–Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $\alpha $, and its tail-index, which is controlled by a parameter $\beta $. In fact, we obtain the classical $1/\sqrt {n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $\alpha \beta >3$ or $\alpha \beta >4$ in the case of Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely on a new second-order Poincaré inequality on the Poisson space, which we derive through a combination of Stein’s method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].

Enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry

The purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poisson-compatible spaces, for stress and displacement variables, and/or on enriched Stokes-compatible space configurations, for the choice of rotation spaces used to weakly enforce stress symmetry. Accordingly, the stress space has to be adapted to ensure stability. Such enrichment procedures are done via space increments with extra bubble functions, which have their support on a single element (in the case of H1-conforming approximations) or with vanishing normal components over element edges (in the case of H(div)-conforming spaces). The advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed, in a way that the number of equations to be solved and the matrix structure are not affected. Enhanced approximations are observed when using the resulting enriched space configurations, which may have different orders of accuracy for the different variables. A general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables, individually. The use of enriched Poisson spaces improves the rates of convergence of stress divergence and displacement variables. Stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order, reaching the maximum rate given by the normal traces (which are not affected).

Functional inequalities for marked point processes

In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincaré inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the law of marked temporal point processes with a Papangelou conditional intensity, and extends a related inequality which is known to hold on a general Poisson space. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Our results are shown to apply to classes of renewal, nonlinear Hawkes and Cox point processes.

The Gamma Stein equation and noncentral de Jong theorems

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate $U$-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (J. Multivariate Anal. 34 (1990) 275–289), Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985) and Dobler and Peccati (Electron. J. Probab. 22 (2017) no. 2), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thale (ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 525–560) and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (Probab. Theory Related Fields 145 (2009) 75–118). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

The fourth moment theorem on the Poisson space

We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result—that has been elusive for several years—shows that the so-called ‘fourth moment phenomenon’, first discovered by Nualart and Peccati [Ann. Probab. 33 (2005) 177–193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carre-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [Ann. Probab. 40 (2012) 2439–2459] and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341–2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

The Twofold Role of Observables in Classical and Quantum Kinematics

Observables have a dual nature in both classical and quantum kinematics: they are at the same time \emph{quantities}, allowing to separate states by means of their numerical values, and \emph{generators of transformations}, establishing relations between different states. In this work, we show how this two-fold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look at the algebraic description of both classical and quantum observables in terms of Jordan-Lie algebras and show how the two algebraic structures are the precise mathematical manifestation of the two-fold role of observables. Then, we turn to the geometric reformulation of quantum kinematics in terms of K\"ahler manifolds. A key achievement of this reformulation is to show that the two-fold role of observables is the constitutive ingredient defining what an observable is. Moreover, it points to the fact that, from the restricted point of view of the transformational role of observables, classical and quantum kinematics behave in exactly the same way. Finally, we present Landsman's general framework of Poisson spaces with transition probability, which highlights with unmatched clarity that the crucial difference between the two kinematics lies in the way the two roles of observables are related to each other.

Fourth moment theorems on the Poisson space: analytic statements via product formulae

We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carre-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Dobler and Peccati (2018) and Dobler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.