本文对Wasserstein曲率或$Γ$ -曲率有下界的连续时间马尔可夫链，给出了新的Poisson型偏差不等式.虽然这两个曲率对于黎曼流形上的布朗运动是等价的，但它们在离散情况下是不可比较的，并且产生不同的偏差界。在生灭过程的情形下，我们给出了相应的生成元的转移率使这类曲率有界的条件，并推广了[Ané，C.和Ledoux，M.关于图上连续时间随机游动的对数Sobolev不等式。可能吧Theory Related Fields 116（2000）573--602]，用于连续时间随机游动，被看作零曲率中的模型。这些尾部估计的一些应用给出了布朗驱动的Ornstein-Uhlenbeck过程和$M/M/1$队列。
英文摘要：
In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or $Γ$-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth--death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [Ané, C. and Ledoux, M. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573--602] for continuous-time random walks, seen as models in null curvature. Some applications of these tail estimates are given for Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues.
By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincaré and weak Poincaré inequalities are established, while another Poincaré type inequality is disproved under some reasonable assumptions.
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.
Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic inde…
We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a powerful logarithmic Sobolev inequality proved by L. Wu (2000), as well as on several variations of the so-called Herbst argument. We provide several applications, in particular to edge counting and more general length power functionals in random geometric graphs, as well as to the convex distance for random point measures recently introduced by M. Reitzner (2…
Let $S$ and $X$ be independent random variables, assuming values in the set of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and $\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We establish a sufficient condition for the tail probability $\mathbb{P}(S\ge \mathbb{E}(S))$ to be larger than $\mathbb{P}(S+X\ge \mathbb{E}(S+X))$. We also apply this result to sums of independent binomial and Poisson random variables.
We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with the causal boundary of the underlying manifold.
Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb{R}^d$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a compact window is derived. Assuming convexity of the typical cylinder base and isotropy of $Z$ a concentration inequality for intrinsic volumes of arbitrary order is established. A number of special cases are discussed, for example the case when the cylinder bases arise from a random rotation of a fixed convex body. Also the situation of expanding windows is st…
In this article, we provide an extension of the Chen-Stein inequality for Poisson approximation in the total variation distance for sums of independent Bernoulli random variables in two ways. We prove that we can improve the rate of convergence (hence the quality of the approximation) by using explicitly constructed signed or positive probability measures, and that we can extend the setting to possibly dependent random variables. The framework which allows this is that of mod-Poisson convergence, and more precisely those mod-Poisson convergent sequences whose residue functions can be expressed…
We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincaré inequality, as well as on the use of Malliavin operators, of Stein's method, and of an (integrated) Mehler's formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order differential operators, and have consequently a clear geometric in…
We establish new explicit bounds on the Gaussian approximation of Poisson functionals based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derive bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators $\unicode{x2014}$ thereby extending the results from (Last, Peccati and Schulte, 2016). Our applications include a CLT for the Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also apply our techniques to de…
Given a vector $F=(F_1,\dots,F_m)$ of Poisson functionals $F_1,\dots,F_m$, we investigate the proximity between $F$ and an $m$-dimensional centered Gaussian random vector $N_Σ$ with covariance matrix $Σ\in\mathbb{R}^{m\times m}$. Apart from finding proximity bounds for the $d_2$- and $d_3$-distances, based on classes of smooth test functions, we obtain proximity bounds for the $d_{convex}$-distance, based on the less tractable test functions comprised of indicators of convex sets. The bounds for all three distances are shown to be of the same order, which is presumably optimal. The bounds are …
Upper bounds for the probabilities $\mathbb{P}(F\geq \mathbb{E} F + r)$ and $\mathbb{P}(F\leq \mathbb{E} F - r)$ are proved, where $F$ is a certain component count associated with a random geometric graph built over a Poisson point process on $\mathbb{R}^d$. The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay. For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in set…
Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$ is either a median or the expectation of a subgraph count $N$. The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure. The tail estimates for subgraph counts…
We derive concentration inequalities for maxima of empirical processes associated with Poisson point processes. The proofs are based on a careful application of Ledoux's entropy method. We demonstrate the utility of the obtained concentration inequalities by application to adaptive intensity estimation.
We consider a Boolean model $Z$ driven by a Poisson particle process $η$ on a metric space $\mathbb{Y}$. We study the random variable $ρ(Z)$, where $ρ$ is a (deterministic) measure on $\mathbb{Y}$. Due to the interaction of overlapping particles, the distribution of $ρ(Z)$ cannot be described explicitly. In this note we derive concentration inequalities for $ρ(Z)$. To this end we first prove two concentration inequalities for functions of a Poisson process on a general phase space.
This note aims at presenting several new theoretical results for the compound Poisson point process, which follows the work of Zhang \emph{et al.} [Insurance~Math.~Econom.~59(2014), 325-336]. The first part provides a new characterization for a discrete compound Poisson point process (proposed by {Acz{é}l} [Acta~Math.~Hungar.~3(3)(1952), 219-224]), it extends the characterization of the Poisson point process given by Copeland and Regan [Ann.~Math.~(1936): 357-362]. Next, we derive some concentration inequalities for discrete compound Poisson point process (negative binomial random variable wit…
We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances b…
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this curvature is nonnegative, its harmonic mean is a lower bound for the spectral gap.
For an ergodic Brownian diffusion with invariant measure $ν$, we consider a sequence of empirical distributions ($ν$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($γ$n) n$\ge$1 along an adapted regular enough class of test functions f such that f --$ν$(f) is a coboundary of the infinitesimal generator A. Denote by $σ$ the diffusion coefficient and $Φ$ the solution of the Poisson equation A$Φ$ = f -- $ν$(f). When the square norm of |$σ$ * $Φ$| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizati…
Convex analysis is fundamental to proving inequalities that have a wide variety of applications in economics and mathematics. In this paper we provide Jensen-type inequalities for functions that are, intuitively, "very" convex. These inequalities are simple to apply and can be used to generalize and extend previous results or to derive new results. We apply our inequalities to quantify the notion "more risk averse" provided in \cite{pratt1978risk}. We also apply our results in other applications from different fields, including risk measures, Poisson approximation, moment generating functions,…
In this paper, we determine the Poisson boundary of the relativistic Brownian motion in two classes of Lorentzian manifolds, namely model manifolds of constant scalar curvature and Robertson--Walker space-times, the latter constituting a large family of curved manifolds. Our objective is two fold: on the one hand, to understand the interplay between the geometry at infinity of these manifolds and the asymptotics of random sample paths, in particular to compare the stochastic compactification given by the Poisson boundary to classical purely geometric compactifications such as the conformal or …
We study the influence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence …
