Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited

Bachmann, Sascha; Peccati, Giovanni

doi:10.1214/16-ejp4235

Cited by 24 publications

(53 citation statements)

References 43 publications

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“…We have s 0 (u) du ≥ 0 since the contrary would lead to P(F −E[F] ≥ 0) ≤ 0 which is obviously wrong. Since inf 0<θ<1 (1 − θ) −1 = 1, we obtain that Our next result was motivated by a question in [1] whether the Mecke formula (cf. [13]) can be combined with the covariance identity to yield reasonable concentration inequalities.…”

Section: Introductionmentioning

confidence: 86%

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Concentration inequalities for measures of a Boolean model

Gieringer¹,

Last²

2018

ALEA

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We consider a Boolean model Z driven by a Poisson particle process η on a metric space Y. We study the random variable ρ(Z), where ρ is a (deterministic) measure on 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.2000 Mathematics Subject Classification. 60D05, 60G55.1 reason for the absence of such formulae is the interaction between the particles from η caused by overlapping. One way out are moment formulae and central limit theorems; see e.g. [10] and [13, Chapter 22]. In this paper we will prove concentration inequalities of the formwhere the function : [0, ∞) → [0, ∞] is determined by Λ and ρ. In the stationary Euclidean case such inequalities were first proved in [7]. Our bounds improve these results. Moreover, we generalize the setting of [7] in several ways. First, we study the Boolean model on a metric space Y and not only on R d . Second, we will allow that compact subsets of Y are intersected by infinitely many Poisson particles. Hence, in general, the random set Z is not closed and its boundary might have fractal properties. Roughly speaking, this means that we can allow for a σ-finite distribution of the typical grain. Closely related models of this type were introduced in [21], a seminal paper on fractal percolation, that was almost completely ignored in the later literature. Third, we consider general measures and not only the volume. Finally, our method allows to treat also Lipschitz functions of these measures.Similarly as in [8,9] our approach is based on a covariance identity for square integrable Poisson functionals. In fact we first prove a concentration inequality for functions of a Poisson process on a general phase space. Using the log-Sobolev inequality, related concentration inequalities were derived in [2,1,19]. Concentration of Poisson functionalsLet (X, X) be a measurable space and let Λ be a σ-finite measure on X. Let η be a Poisson process on X with intensity measure Λ, defined over a probability space (Ω, A, P); see [13]. In particular, η is a point process, that is a measurable mapping from Ω to the space N = N(X) of all σ-finite measures with values inN 0 := {∞, 0, 1, 2, . . .}, where N is equipped with the smallest σ-field N such that µ → µ(B) is measurable for all B ∈ X. The distribution of η is denoted by Π Λ := P(η ∈ ·). Since we are only interested in distributional properties of η, Corollary 6.5 in [13] shows that it is no restriction of generality to assume that η is proper. This means that there exist random elements X 1 , X 2 , . . . in X and anN 0 -valued random variable κ such that almost surely η = κ n=1 δ X n . Let 0 ≤ t ≤ 1 and Y 1 , Y 2 , . . . be a sequence of independent random variables with distribution (1 − t)δ 0 + tδ 1 , independent of η. Define η t := κ n=1 Y n δ X n as the t-thinning of η. Then η t and η − η t are independ...

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Section: Introductionmentioning

confidence: 86%

“…At least in the case of bounded grains, this application already improves the correspondent result of [7]. However, the functionals considered in [1] appear unable to incorporate the volume fraction. To be more precise, in the setting of Corollary 4.3, the bound (4.6) is superior to the result…”

Section: Stationary Boolean Modelsmentioning

confidence: 99%

Concentration inequalities for measures of a Boolean model

Gieringer¹,

Last²

2018

ALEA

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“…The latter paper provides several refinements of a method for proving tail estimates for Poisson functionals (also known as the entropy-method), which is based on (modified) logarithmic Sobolev inequalities, and which was particularly studied in the seminal work by Wu [28], extending previous findings by Ané, Bobkov and Ledoux [1,5]. Combining Wu's modified logarithmic Sobolev inequality with the famous Mecke formula for Poisson processes, the authors of [2] were able to adapt concentration techniques for product space functionals, which were particularly developed by Boucheron, Lugosi and Massart [8], and also by Maurer [22], to the setting of Poisson processes. This approach adds a lot of flexibility to the entropy-method, and a remarkable feature of the obtained techniques is that they allow to deal with functionals build over Poisson processes with infinite intensity measure.…”

Section: Introductionmentioning

confidence: 88%

“…Inequalities of this type are usually called concentration inequalities. In order to derive our estimates, we use and enhance a method that was recently developed by S. Bachmann and G. Peccati in [2]. The latter paper provides several refinements of a method for proving tail estimates for Poisson functionals (also known as the entropy-method), which is based on (modified) logarithmic Sobolev inequalities, and which was particularly studied in the seminal work by Wu [28], extending previous findings by Ané, Bobkov and Ledoux [1,5].…”

Section: Introductionmentioning

confidence: 99%

“…First applications for these techniques are worked out in [2] and also in [3], where concentration bounds for certain Poisson U-statistics with positive kernels are established. A crucial property that was exploited in the latter investigations is that adding a point to the Poisson process cannot decrease the value of the considered functionals.…”

Section: Introductionmentioning

confidence: 99%

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Concentration for Poisson functionals: Component counts in random geometric graphs

Bachmann

2016

Stochastic Processes and their Applications

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Upper bounds for the probabilities P(F ≥ EF + r) and P(F ≤ EF − r) are proved, where F is a certain component count associated with a random geometric graph built over a Poisson point process on 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 settings where the underlying Poisson process has infinite intensity measure.MSC: primary 60D05; secondary 05C80, 60C05

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Limit theory for the Gilbert graph

Reitzner

Schulte

Thäle

2017

Advances in Applied Mathematics

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For a given homogeneous Poisson point process in R d two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance.

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Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited

Cited by 24 publications

References 43 publications

Concentration inequalities for measures of a Boolean model

Concentration inequalities for measures of a Boolean model

Concentration for Poisson functionals: Component counts in random geometric graphs

Limit theory for the Gilbert graph

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