Mixing properties and central limit theorem for associated point processes

Poinas, Arnaud; Delyon, Bernard; Lavancier, Frédéric

doi:10.48550/arxiv.1705.02276

Cited by 5 publications

(11 citation statements)

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“…Using our Theorem 3.3, we will prove an analogous characterization (by finite dimensional distributions) of negative association for random measures on Polish spaces (Theorem 4.4). Similar results on negative association for point processes on R d and on locally compact Polish spaces has been recently given by Poinas et al [47,Theorem 2.3] and Lyons [40, paragraph 3.7] respectively. Though the latter result is in the context of determinantal point processes, the proof applies to general negatively associated point processes.…”

Section: Introductionsupporting

confidence: 84%

“…The following definition is an extension to random measures of definitions used by Lyons [40] and Poinas et al [47] for point processes. Definition 4.2.…”

Section: Association Of Random Measuresmentioning

confidence: 99%

“…For negative association this property does not hold. In Poinas et al [47,Theorem 2.3] the proof of an analog of Theorem 4.4 is given for point processes on S = R d . They use a variant of the monotone class theorem.…”

Section: Association Of Random Measuresmentioning

confidence: 99%

“…Association of real random fields on Z d and R d were used to obtain central limit theorems for random fields, see e.g. Bulinski and Shashkin [7], Bulinski and Spodarev [8] or Poinas et al [47] and references therein. We shall give examples of associated (positive and negative) random fields later in Section 5 and in the Appendix.…”

Section: Introductionmentioning

confidence: 99%

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Some remarks on associated random fields, random measures and point processes

Szekli

Yogeshwaran

2019

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In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space with a closed partial order (POP space), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in [38]. We use these results to show on POP spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in [47] and [40] which restrict to point processes in R d and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.

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Section: Introductionsupporting

confidence: 84%

“…The following definition is an extension to random measures of definitions used by Lyons [40] and Poinas et al [47] for point processes. Definition 4.2.…”

Section: Association Of Random Measuresmentioning

confidence: 99%

Section: Association Of Random Measuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some remarks on associated random fields, random measures and point processes

Szekli

Yogeshwaran

2019

Preprint

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“…We now state an important concentration inequality for determinantal point processes as well as a bound for the α p,q mixing coefficient. The next two results can be found as Theorem 3.6 in [29] and Corollary 4.2 in [32] respectively. Theorem 12.2.…”

Section: Discussionmentioning

confidence: 92%

Hyperuniform and rigid stable matchings

Klatt

Yogeshwaran

2018

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We study a stable partial matching τ of the (possibly randomized) d-dimensional lattice with a stationary determinantal point process Ψ on R d with intensity α > 1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψ τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ τ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so-called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behaviour. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Further, if Ψ is a Poisson process then Ψ τ has an exponentially decreasing truncated pair correlation function.

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