The Stein Method | ScienceGate

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

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AN IDENTIFICATION PROBLEM IN AN URN AND BALL MODEL WITH HEAVY TAILED DISTRIBUTIONS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990143 ◽

2009 ◽

Vol 24 (1) ◽

pp. 77-97

Author(s):

Christine Fricker ◽

Fabrice Guillemin ◽

Philippe Robert

Keyword(s):

Basic Problem ◽

Identification Problem ◽

Total Variation Norm ◽

Heavy Tailed Distributions ◽

Ball Problem ◽

Original Number ◽

Explicit Bounds ◽

Variation Norm ◽

Heavy Tailed ◽

Stein Method

We consider in this article an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are independent and identically distributed random variables with a heavy tailed probability distribution—for instance a Pareto or a Weibull distribution. We draw a small fraction p≪1 of the total number of balls. The basic problem addressed in this article is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and the Chen–Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. Finally, we establish explicit bounds between the two distributions when each ball is drawn with fixed probability p.

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Comments on “A critique of the Ayres-Stein method of predicting cleavage planes in metals”

Scripta Metallurgica ◽

10.1016/0036-9748(78)90006-6 ◽

1978 ◽

Vol 12 (11) ◽

pp. 979-982 ◽

Cited By ~ 3

Author(s):

R.A. Ayres ◽

J.E. Hack ◽

D.F. Stein

Keyword(s):

Stein Method

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Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

Advances in Applied Probability ◽

10.1017/s0001867800001634 ◽

2007 ◽

Vol 39 (01) ◽

pp. 128-140 ◽

Cited By ~ 1

Author(s):

Etienne Roquain ◽

Sophie Schbath

Keyword(s):

Markov Chain ◽

Poisson Distribution ◽

Approximation Error ◽

Rare Event ◽

Poisson Approximation ◽

Rare Word ◽

Compound Poisson ◽

Compound Poisson Approximation ◽

Stein Method ◽

Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

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A Phase Transition for the Distribution of Matching Blocks

Combinatorics Probability Computing ◽

10.1017/s0963548300001930 ◽

1996 ◽

Vol 5 (2) ◽

pp. 139-159 ◽

Cited By ~ 2

Author(s):

Claudia Neuhauser

Keyword(s):

Phase Transition ◽

Law Of Large Numbers ◽

Poisson Approximation ◽

Finite Alphabet ◽

Strong Law ◽

Large Numbers ◽

Stein Method

We show distributional results for the length of the longest matching consecutive subsequence between two independent sequences A1, A2, …, Am and B1, B2, …, Bn whose letters are taken from a finite alphabet. We assume that A1, A2, … are i.i.d. with distribution μ and B1, B2, … are i.i.d. with distribution ν. It is known that if μ and v are not too different, the Chen–Stein method for Poisson approximation can be used to establish distributional results. We extend these results beyond the region where the Chen–Stein method was previously successful. We use a combination of ‘matching by patterns’ results obtained by Arratia and Waterman [1], and the Chen–Stein method to show that the Poisson approximation can be extended. Our method explains how the matching is achieved. This provides an explanation for the formulas in Arratia and Waterman [1] and thus answers one of the questions posed in comment F19 in Aldous [2]. Furthermore, in the case where the alphabet consists of only two letters, the phase transition observed by Arratia and Waterman [1] for the strong law of large numbers extends to the distributional result. We conjecture that this phase transition on the distributional level holds for any finite alphabet.

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Poisson Approximation and the Chen-Stein Method

Statistical Science ◽

10.1214/ss/1177012015 ◽

1990 ◽

Vol 5 (4) ◽

pp. 403-424 ◽

Cited By ~ 185

Author(s):

Richard Arratia ◽

Larry Goldstein ◽

Louis Gordon

Keyword(s):

Poisson Approximation ◽

Stein Method

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A non-increasing tree growth process for recursive trees and applications

Combinatorics Probability Computing ◽

10.1017/s0963548320000073 ◽

2020 ◽

pp. 1-26

Author(s):

Laura Eslava

Keyword(s):

Tree Growth ◽

Degree Distribution ◽

Growth Process ◽

Convergence Rates ◽

Positive Probability ◽

New Process ◽

Empty Subset ◽

Recursive Trees ◽

Stein Method ◽

Kingman’S Coalescent

Abstract We introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where T n is a rooted labelled tree on n vertices and σ n is a permutation of the vertex labels. The construction of (T n , σ n ) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ T n with positive probability. The key feature of the process is that the shape of T n has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process. We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.

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On asymptotic properties of the distribution of the number of pairs of H-connected tuples

Discrete Mathematics and Applications ◽

10.1515/dma-2002-0407 ◽

2002 ◽

Vol 12 (4) ◽

Author(s):

V. G. Mikhailov

Keyword(s):

Asymptotic Properties ◽

Sufficient Conditions ◽

Basic Research ◽

Poisson Approximation ◽

Explicit Estimate ◽

Research Grants ◽

Local Variant ◽

Sufficient Conditions For Convergence ◽

Stein Method ◽

Independent Identically Distributed

AbstractThe main result of this paper is a theorem about convergence of the distribution of the number of pairs of H-connected s-tuples in two independent sequences of independent identically distributed variables. The concept of H-connection is a generalisation of the concept of H-equivalence of tuples. We give sufficient conditions for convergence and an explicit estimate of the rate of convergence. We use the local variant of the Chen-Stein method for estimating the accuracy of Poisson approximation for distribution of the set of dependent random indicators. The main results of this paper were announced in [7].The research was supported by the Russian Foundation for Basic Research, grants 02-01-00266 and 00-15-96136.

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