The limit distribution of the maximum probability nearest-neighbour ball

2019 ◽  
Vol 56 (2) ◽  
pp. 574-589
Author(s):  
László Györfi ◽  
Norbert Henze ◽  
Harro Walk
Keyword(s):  
Limit Theorem ◽  
Probability Measure ◽  
Limit Distribution ◽  
Upper Bound ◽  
Extreme Value Distribution ◽  
Nearest Neighbour ◽  
Maximum Probability ◽  
Absolutely Continuous ◽  
Mild Conditions ◽  
Poisson Limit

AbstractLet X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

scholarly journals The local limit theorem for sums of random variables defined on a homogeneous Markov chain in the case of the stable limit distribution law

1961 ◽  
Vol 1 (1-2) ◽  
pp. 7-16
Author(s):  
A. Aleškevičienė
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Distribution ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Homogeneous Markov Chain ◽  
Stable Limit ◽  
Distribution Law ◽  
Sums Of Random Variables ◽  
Stable Limit Distribution

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: A. Алешкявичене. Локальная предельная теорема для сумм случайных величин, связанных в однородную цепь Маркова в случае устойчивого предельного распределения A. Aleškevičienė. Lokalinė ribinė teorema atsitiktinių dydžių, surištų homogenine Markovo grandine, sumoms stabilaus ribinio dėsnio atveju  

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

Exponential Boundedness and Amenability of Open Subsemigroups of Locally Compact Groups

1994 ◽  
Vol 46 (06) ◽  
pp. 1263-1274 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Open Subset ◽  
Compact Support ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Complete Proof ◽  
Absolutely Continuous ◽  
Crucial Part

Abstract Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L ∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

2004 ◽  
Vol 41 (01) ◽  
pp. 83-92 ◽  
Author(s):  
Jean Bérard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Space Time ◽  
Nearest Neighbour ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Almost All

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

A Poisson limit theorem for incomplete symmetric statistics

1983 ◽  
Vol 20 (01) ◽  
pp. 47-60 ◽  
Author(s):  
M. Berman ◽  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Poisson Limit ◽  
Symmetric Statistics ◽  
Limit Result ◽  
Independent Identically Distributed

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

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