A note on gaps in proofs of central limit theorems

Biscio, Christophe Ange Napoléon; Poinas, Arnaud; Waagepetersen, Rasmus

doi:10.1016/j.spl.2017.11.009

Cited by 9 publications

(8 citation statements)

References 9 publications

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“…By Lemma F5, sup n∈N ES 2 n < ∞. Thus, by Lemma F2 in Bolthausen (1982; see also the discussion in Biscio, Poinas, & Waagepetersen, 2018), Theorem A1 is proved if…”

Section: Proof Of Theorem A1mentioning

confidence: 66%

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A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Biscio

Waagepetersen

2019

Scandinavian J Statistics

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We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.

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“…By Lemma F5, sup n∈N ES 2 n < ∞. Thus, by Lemma F2 in Bolthausen (1982; see also the discussion in Biscio, Poinas, & Waagepetersen, 2018), Theorem A1 is proved if…”

Section: Proof Of Theorem A1mentioning

confidence: 66%

“…Lemma F6 (Biscio et al, 2018). Let {X n } n∈N be a sequence of random variables in R p , for p ∈ N, such that…”

Section: Appendix F Lemmasmentioning

confidence: 99%

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Biscio

Waagepetersen

2019

Scandinavian J Statistics

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“…The question of asymptotic normality of ( 24) is a more delicate matter, however, and one would likely need to consider some notion of mixing for the FMPP in question (cf. Biscio et al 2018;Biscio and Waagepetersen 2019).…”

Section: Definitionmentioning

confidence: 99%

Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data

Ghorbani

Cronie

Mateu

et al. 2020

TEST

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This paper treats functional marked point processes (FMPPs), which are defined as marked point processes where the marks are random elements in some (Polish) function space. Such marks may represent, for example, spatial paths or functions of time. To be able to consider, for example, multivariate FMPPs, we also attach an additional, Euclidean, mark to each point. We indicate how the FMPP framework quite naturally connects the point process framework with both the functional data analysis framework and the geostatistical framework. We further show that various existing stochastic models fit well into the FMPP framework. To be able to carry out nonparametric statistical analyses for FMPPs, we study characteristics such as product densities and Palm distributions, which are the building blocks for many summary statistics. We proceed to defining a new family of summary statistics, so-called weighted marked reduced moment measures, together with their nonparametric estimators, in order to study features of the functional marks. We further show how other summary statistics may be obtained as special cases of these summary statistics. We finally apply these tools to analyse population structures, such as demographic evolution and sex ratio over time, in Spanish provinces.

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“…We then only need to prove the asymptotic normality of s S n . Moreover, since sup n Er s S 2 n s ă 8 then this will be a consequence of the following condition (see [5,7]) lim nÑ8 Erpiλ´s S n q exppiλ s S n qs " 0, @λ P R.…”

Section: First Step: Bounded Variables Without Loss Of Generality Wmentioning

confidence: 99%

“…Since Σ n does not necessary converge, this result is not a direct application of the Cramér-Wold device. Instead, a detailed proof is given in [5].…”

Section: Central Limit Theorem For Functionals Of Dppsmentioning

confidence: 99%

Mixing properties and central limit theorem for associated point processes

Poinas¹,

Delyon²,

Lavancier³

2019

Bernoulli

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Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove α-mixing properties for associated spatial point processes by controlling their α-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the α-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.We then establish in Section 3 a general central limit theorem (CLT) for random fields defined as a function of an associated point process (Theorem 3.1). A standard method for proving this kind of theorem is to rely on sufficiently fast decaying α-mixing coefficients along with some moment assumptions. We use an alternative procedure that exploits both the mixing properties and the association property. This results in weaker assumptions on the underlying point process, that can have slower decaying mixing coefficients. This improvement allows in particular to include all standard DPPs, some of them being otherwise excluded with the first approach (like for instance DPPs associated to the Bessel-type kernels [4]).Section 4 discusses in detail the case of DPPs, where we derive a tight explicit bound for their α-mixing coefficients and prove a central limit theorem for certain functionals of a DPP (Theorem 4.4). Specifically, these functionals write as a sum of a bounded function of the DPP, over subsets of close-enough points of the DPP. A particular case concerns sums over p-tuple of close enough points of the DPP, which are frequently encountered in asymptotic inference. Limit theorems in this setting have been established in [38] when p " 1, and in [3] for stationary DPPs and p ě 1. We thus extend these studies to sums over any subsets and without the stationary assumption. As a statistical application, we consider the parametric estimation of second-order intensity reweighted stationary DPPs. These DPPs have an inhomogeneous first order intensity, but translation-invariant higher order (reweighted) intensities. We prove that the two-step estimator introduced in [39], designed for this kind of inhomogeneous point process models, is consistent and asymptotically normal when applied to DPPs. Associated point processes and α-mixing NotationIn this paper, we consider locally finite simple point processes on R d , for a fixed d P N. Some theoretical background on point processes can be found in [12,34]. We denote by Ω the set of locally finite point configurations in R d . For X P Ω and A Ă R d , we write N pAq :" cardpX ∩ Aq for the random variable representing the number of points of X that fall in A. W...

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A note on gaps in proofs of central limit theorems

Cited by 9 publications

References 9 publications

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data

Mixing properties and central limit theorem for associated point processes

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