Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal Distribution

Sunklodas, J.

doi:10.1007/978-3-662-04172-7_3

Cited by 18 publications

(10 citation statements)

References 155 publications

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“…Proof of Theorem 1.1. We need the following central limit theorem appearing in a survey article by Sunklodas [14]: THEOREM 10.1 (In English translation [13], page 133, Theorem 10). Let X = L i=1 X i where X 1 , .…”

Section: Proof Of Theorem 12 Let X Denote Either N or Amentioning

confidence: 99%

Central limit theorems for random polygons in an arbitrary convex set

Pardon¹

2011

Ann. Probab.

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We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets $K\subset\mathbb{R}^2$ without imposing any regularity conditions on the boundary $\partial K$. Our main result is a central limit theorem for both the area and the number of vertices, settling a well-known conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP568 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Proof Of Theorem 12 Let X Denote Either N or Amentioning

confidence: 99%

Central limit theorems for random polygons in an arbitrary convex set

Pardon¹

2011

Ann. Probab.

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“…In particular, we replace the moments in the estimates by almost sure bounds. [29] for an excellent overview. We point out Kolmogorov distance approximations such as Stein [30] and Tikhomirov [32], and Edgeworth expansions and local limit CLTs such as Götze and Hipp [18,19] and Heinrich [20,21].…”

Section: Applicationsmentioning

confidence: 99%

“…The accuracy of approximation is most often measured by the Kolmogorov and Kantorovich metrics (see Sunklodas [29] for an overview with many references).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, these results are extended to locally dependent summands and Section 4 shows their application to some simple examples. These applications are instances of m-dependent sequences and finite-range random fields for which a large amount of literature is available (see again Sunklodas [29] for references). We would like to point out the local limit CLTs and Edgeworth expansions in Götze and Hipp [18,19]) and Heinrich [20,21], which also cover the examples in Section 4.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric and centered binomial approximation of sums of locally dependent random variables

Roellin

2008

Electron. J. Probab.

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Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric binomial distribution, serving as a natural alternative to the normal distribution in discrete settings. The bounds are given with respect to the total variation and a local limit metric. Under appropriate smoothness properties of the summands, the same order of accuracy as in the Berry-Essen Theorem is achieved. The approximation of the total number of points of a point processes is also considered. The results are applied to the exceedances of the r-scans process and to the Matérn hardcore point process type I to obtain explicit bounds with respect to the two metrics.

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“…Even though the afore-cited applications are important contributions of our article as well as crucial motivations for our work, our main contribution can be said to lie within the realm of limit theorems for asymptotically independent random fields as in [87,25,18,45,20,8]. We shall elaborately survey this literature to contextualise our results.…”

Section: Introductionmentioning

confidence: 99%

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

2018

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Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of α-mixing (for local statistics) and exponential α-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.Keywords Clustering spin models · mixing random fields · central limit theorem · Cayley graphs · fast decaying covariance · Gaussian random field · determinantal point process · exponentially quasi-local statistics · cubical complexes · nearest-neighbour graphs Mathematics Subject Classification (2000) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs · 60G60 Random fields · 60F05 Central limit theorems and other weak theorems · 60D05 Geometric probability and stochastic geometry.

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Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal Distribution

Cited by 18 publications

References 155 publications

Central limit theorems for random polygons in an arbitrary convex set

Central limit theorems for random polygons in an arbitrary convex set

Symmetric and centered binomial approximation of sums of locally dependent random variables

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

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