Approximation Methods in Probability Theory

Čekanavičius, Vydas

doi:10.1007/978-3-319-34072-2

Cited by 28 publications

(35 citation statements)

References 124 publications

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“…Lemma 4.2 contains well-known properties of Levy's concentration function, see, for example, Chapter 1 in [19] or Section 1.5 in [5].…”

Section: Auxiliary Resultsmentioning

confidence: 99%

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On closeness of two discrete weighted sums

Čekanavičius¹,

Vellaisamy²

2018

Modern Stochastics: Theory and Applications

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The effect that weighted summands have on each other in approximations of S = w1S1 + w2S2 + · · · + wN SN is investigated. Here, Si's are sums of integer-valued random variables, and wi denote weights, i = 1, . . . , N . Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the Si has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.

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“…Lemma 4.2 contains well-known properties of Levy's concentration function, see, for example, Chapter 1 in [19] or Section 1.5 in [5].…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…Lemma 4.6 is a well-known inversion inequality for lattice distributions. Its proof can be found, for example, in [5], Lemma 5.1.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

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On closeness of two discrete weighted sums

Čekanavičius¹,

Vellaisamy²

2018

Modern Stochastics: Theory and Applications

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“…Lemma 3.5 is a well known inversion inequality for lattice distributions. Its proof can be found, for example, in [5], Lemma 5.1.…”

Section: Lemma 33 Let the Conditions Stated In (2)-(3) Be Satisfiedmentioning

confidence: 99%

On large deviations for sums of discrete m-dependent random variables

Čekanavičius

Vellaisamy

2019

Stochastics

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The ratio P (S n = x)/P (Z n = x) is investigated for three cases: (a) when S n is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and Z n is Poisson rv; (b) when S n is a statistic of 2-runs and Z n is negative binomial rv; and (c) when S n is statistic of N (1, 1)-events and Z n is a binomial r.v. We also consider the approximation of P (S n x) by Poisson distribution with parameter depending on x.

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“…Moreover, H 3 = L(T − ∆), with ∆ = n i=1 a i (ν i − 1). There exists a lot of results on the compound Poisson approximation of the distributions n i=1 F i , see Arak and Zaitsev [2], Barbour and Chryssaphinou [3], Zaitsev [23], Roos [18], Cekanavičius [6] and the bibliography therein. However, most of these results require that distributions F i are appropriately centered.…”

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confidence: 99%

Rare Events and Poisson Point Processes

Götze

Зайцев

2020

J Math Sci

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The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent terms by the accompanying compound Poisson laws may be interpreted as rather sharp quantitative estimates for the closeness between the sample containing independent observations of rare events and the Poisson point process which is obtained after a Poissonization of the initial sample.

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Approximation Methods in Probability Theory

Cited by 28 publications

References 124 publications

On closeness of two discrete weighted sums

On closeness of two discrete weighted sums

On large deviations for sums of discrete m-dependent random variables

Rare Events and Poisson Point Processes

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