Examples of Stationary Sequences with Approximate Negative Dependence

2019 ◽  
pp. 305-318
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Spectral Density ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Stationary Processes ◽  
Negative Dependence ◽  
Stationary Sequences ◽  
Determinantal Point Process ◽  
Dependent Random Variables ◽  
Associated Sequences

In this chapter, we treat several examples of stationary processes which are asymptotically negatively dependent and for which the results of Chapter 9 apply. Many systems in nature are complex, consisting of the contributions of several independent components. Our first examples are functions of two independent sequences, one negatively dependent and one interlaced mixing. For instance, the class of asymptotic negatively dependent random variables is used to treat functions of a determinantal point process and a Gaussian process with a positive continuous spectral density. Another example is point processes based on asymptotically negatively or positively associated sequences and displaced according to a Gaussian sequence with a positive continuous spectral density. Other examples include exchangeable processes, the weighted empirical process, and the exchangeable determinantal point process.

scholarly journals Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

2021 ◽  
Vol 58 (2) ◽  
pp. 469-483
Author(s):  
Jesper Møller ◽  
Eliza O’Reilly
Keyword(s):  
Finite Number ◽  
Point Process ◽  
Point Processes ◽  
Parametric Models ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
The Difference ◽  
Weaker Conditions ◽  
Palm Distributions

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

scholarly journals High-performance sampling of generic determinantal point processes

2020 ◽  
Vol 378 (2166) ◽  
pp. 20190059 ◽  
Author(s):  
Jack Poulson
Keyword(s):  
Point Process ◽  
Spectral Decomposition ◽  
High Performance ◽  
Point Processes ◽  
Cholesky Factorization ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Decomposition Approach ◽  
Map Inference ◽  
Sampling Schemes

Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975 Adv. Appl. Probab. 7 , 83–122) as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system (Kulesza & Taskar 2012 Found. Trends Mach. Learn. 5 , 123–286). The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive set-up cost. Researchers Launay et al. 2018 ( http://arxiv.org/abs/1802.08429 ); Chen & Zhang 2018 NeurIPS ( https://papers.nips.cc/paper/7805-fast-greedy-map-inference-for-determinantal-point-process-to-improve-recommendation-diversity.pdf ) have begun to connect DPP sampling to LDL H factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. This article proves that trivial modifications of LU and LDL H factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Furthermore, it is experimentally shown that even dynamically scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. The software developed as part of this research, Catamari ( hodgestar.com/catamari ) is released under the Mozilla Public License v.2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

On a point process with independent locations

1975 ◽  
Vol 12 (3) ◽  
pp. 435-446 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Independent Random Variables ◽  
General Conditions

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.

scholarly journals A note on the simulation of the Ginibre point process

2015 ◽  
Vol 52 (4) ◽  
pp. 1003-1012 ◽  
Author(s):  
Laurent Decreusefond ◽  
Ian Flint ◽  
Anais Vergne
Keyword(s):  
Point Process ◽  
Random Matrix ◽  
Point Processes ◽  
Matrix Theory ◽  
Applied Mathematics ◽  
Fixed Number ◽  
Extended Version ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Ginibre Point Process

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

scholarly journals Determinantal Point Process Mixtures Via Spectral Density Approach

2020 ◽  
Vol 15 (1) ◽  
pp. 187-214 ◽  
Author(s):  
Ilaria Bianchini ◽  
Alessandra Guglielmi ◽  
Fernando A. Quintana
Keyword(s):  
Spectral Density ◽  
Point Process ◽  
Determinantal Point Process

On the variance to mean ratio for random variables from Markov chains and point processes

1998 ◽  
Vol 35 (2) ◽  
pp. 303-312 ◽  
Author(s):  
Timothy C. Brown ◽  
Kais Hamza ◽  
Aihua Xia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Point Process ◽  
Continuous Time ◽  
Infinitesimal Generator ◽  
Point Processes ◽  
Random Variables ◽  
Simple Point ◽  
Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

θ-stationary point processes and their second-order analysis

1981 ◽  
Vol 18 (04) ◽  
pp. 864-878
Author(s):  
Karen Byth
Keyword(s):  
Point Process ◽  
Spatial Patterns ◽  
Stationary Point ◽  
Point Processes ◽  
Simulated Data ◽  
Second Order ◽  
Stationary Processes ◽  
Order Analysis ◽  
Stationary Point Processes ◽  
Point Of Reference

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

θ-stationary point processes and their second-order analysis

1981 ◽  
Vol 18 (4) ◽  
pp. 864-878 ◽  
Author(s):  
Karen Byth
Keyword(s):  
Point Process ◽  
Spatial Patterns ◽  
Stationary Point ◽  
Point Processes ◽  
Simulated Data ◽  
Second Order ◽  
Stationary Processes ◽  
Order Analysis ◽  
Stationary Point Processes ◽  
Point Of Reference

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

Preservation of Stochastic Orderings under Random Mapping by Point Processes

1995 ◽  
Vol 9 (4) ◽  
pp. 563-580 ◽  
Author(s):  
Moshe Shaked ◽  
Tityik Wong
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Stochastic Orders ◽  
Stochastic Orderings ◽  
Random Mapping

Let N = [N(t), t ≥ 0] be a point process, and let T1 and T2 be two positive random variables that are independent of N. Define Mk = N(Tk), k = 1,2. In this paper we study conditions on the process N under which some stochastic orders between T1 and T2 are transformed into stochastic orders between M1 and M2. In some cases the converses are also shown to be true. Some applications and examples are given.

scholarly journals Statistical Properties and Applications of Correlated Length Intervals of Some Stationary Random Point Processes

ISRN Optics ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Cherif Bendjaballah
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Finite Interval ◽  
Probability Distributions ◽  
Random Variables ◽  
Statistical Properties ◽  
Time Intervals ◽  
Probability Of Error ◽  
Transmission Of Information ◽  
Theoretical Results

The point process, a sequence of random univariate random variables derived from correlated bivariate random variables as modeled by Arnold and Strauss, has been examined. Statistical properties of the time intervals between the points as well as the probability distributions of the number of points registered in a finite interval have been analyzed specifically in function of the coefficient of correlation. The results have been applied to binary detection and to the transmission of information. Both the probability of error and the cut-off rate have been bounded. Several simulations have been generated to illustrate the theoretical results.

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