scholarly journals On the excursion area of perturbed Gaussian fields

2020 ◽  
Vol 24 ◽  
pp. 252-274
Author(s):  
Elena Di Bernardino ◽  
Anne Estrade ◽  
Maurizia Rossi
Keyword(s):  
Limit Theorem ◽  
Random Perturbation ◽  
Wasserstein Distance ◽  
Expansion Formula ◽  
Kinematic Formula ◽  
Small Perturbations ◽  
Asymptotically Normal ◽  
Excursion Sets ◽  
Growing Domain ◽  
Non Gaussian

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2 under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

scholarly journals DETERMINING SOURCE CUMULANTS IN FEMTOSCOPY WITH GRAM–CHARLIER AND EDGEWORTH SERIES

2011 ◽  
Vol 26 (24) ◽  
pp. 1771-1782 ◽  
Author(s):  
H. C. EGGERS ◽  
M. B. DE KOCK ◽  
J. SCHMIEGEL
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Fourth Order ◽  
Momentum Space ◽  
Crucial Role ◽  
Gaussian Distributions ◽  
Edgeworth Series ◽  
The Central Limit Theorem ◽  
Non Gaussian

Lowest-order cumulants provide important information on the shape of the emission source in femtoscopy. For the simple case of noninteracting identical particles, we show how the fourth-order source cumulant can be determined from measured cumulants in momentum space. The textbook Gram–Charlier series is found to be highly inaccurate, while the related Edgeworth series provides increasingly accurate estimates. Ordering of terms compatible with the Central Limit Theorem appears to play a crucial role even for non-Gaussian distributions.

scholarly journals Excursion sets and non-Gaussian void statistics

2011 ◽  
Vol 83 (2) ◽  
Author(s):  
Guido D’Amico ◽  
Marcello Musso ◽  
Jorge Noreña ◽  
Aseem Paranjape
Keyword(s):  
Excursion Sets ◽  
Non Gaussian

scholarly journals The distribution of m-ary search trees generated by van der Corput sequences

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Wolfgang Steiner
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Function Approach ◽  
Search Trees ◽  
Asymptotically Normal

We study the structure of $m$-ary search trees generated by the van der Corput sequences. The height of the tree is calculated and a generating function approach shows that the distribution of the depths of the nodes is asymptotically normal. Additionally a local limit theorem is derived.

Limit theorems and estimation theory for branching processes with an increasing random number of ancestors

1997 ◽  
Vol 34 (2) ◽  
pp. 309-327 ◽  
Author(s):  
J. P. Dion ◽  
N. M. Yanev
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Number ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Transfer Theorem ◽  
Asymptotically Normal ◽  
The Mean ◽  
Watson Process

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

scholarly journals Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules

10.37236/4914 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Stavros Kousidis ◽  
Ernst Schulte-Geers
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Current Algebra ◽  
Irreducible Representations ◽  
Moody Algebra ◽  
Asymptotically Normal ◽  
Asymptotic Parameter ◽  
Demazure Modules ◽  
Fusion Products

We prove asymptotic normality of the distributions defined by $q$-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of $\frak{sl}_2$. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to $\frak{sl}_2$. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type $A$ standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, $N$ say.

scholarly journals Asymptotics of a Locally Dependent Statistic on Finite Reflection Groups

10.37236/9454 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Frank Röttger
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Coxeter Groups ◽  
Random Permutation ◽  
Wasserstein Distance ◽  
Reflection Groups ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Dependency Structures

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was listed as an interesting direction by Chatterjee and Diaconis (2017). For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme to signed permutations, i.e. elements of Coxeter groups of type $ \mathtt{B}_n $, which are also known as the hyperoctahedral groups.  Furthermore, a similar central limit theorem for elements of Coxeter groups of type $\mathtt{D}_n$ is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups. In addition, we show a two-dimensional central limit theorem via the Cramér-Wold device.

scholarly journals Applying the Fluctuation–Dissipation Theorem to a Two-Layer Model of Quasigeostrophic Turbulence

2015 ◽  
Vol 72 (8) ◽  
pp. 3161-3177 ◽  
Author(s):  
Nicholas J. Lutsko ◽  
Isaac M. Held ◽  
Pablo Zurita-Gotor
Keyword(s):  
Internal Variability ◽  
Empirical Orthogonal Functions ◽  
Unperturbed System ◽  
Orthogonal Functions ◽  
Small Perturbations ◽  
Fluctuation Dissipation ◽  
Fluctuation Dissipation Theorem ◽  
Small Force ◽  
Quasigeostrophic Model ◽  
Non Gaussian

Abstract The fluctuation–dissipation theorem (FDT) provides a means of calculating the response of a dynamical system to a small force by constructing a linear operator that depends only on data from the internal variability of the unperturbed system. Here the FDT is used to estimate the response of a two-layer quasigeostrophic model to two zonally symmetric torques, both barotropic, with the same sign of the forcing in the two layers, and baroclinic, with opposite sign forcing in the two layers. The supercriticality of the model is also varied to test how the FDT fares, as this parameter is varied. To perform the FDT calculations the data are decomposed onto empirical orthogonal functions (EOFs) and only those EOFs that are well resolved are retained in the FDT calculations. In the barotropic case good qualitative estimates are obtained for all values of the supercriticality, though the FDT consistently overestimates the response, perhaps because of significant non-Gaussian behavior present in the model. Nevertheless, this adds to the evidence that the annular-mode time scale plays an important role in determining the response of the midlatitudes to small perturbations. The baroclinic case is more challenging for the FDT. However, by constructing different bases with which to calculate the EOFs, it is shown that the issue in this case is that the baroclinic variability is poorly sampled, not that the FDT fails. The strategies developed in order to generate these estimates may be applicable to situations in which the FDT is applied to larger systems.

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