On approximations to generalized Poisson distributions

Bening, V. E.; Korolev, V. Yu.; Shorgin, Sergey

doi:10.1007/bf02400920

Cited by 20 publications

(8 citation statements)

References 7 publications

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“…where the supremum is taken over all distributions F of the random variable X 1 satisfying conditions (1). It can easily be seen that C(δ) ≥ C * (δ).…”

Section: The Lower Boundsmentioning

confidence: 99%

On the constants in the uniform and non-uniform versions of the Berry-Esseen inequality for Poisson random sums

Nefedova

Shevtsova

2010

International Congress on Ultra Modern Telecommunications and Control Systems

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The sums of large number of independent random variables are very popular mathematical models for many real objects. The central limit theorem states that the distribution of such sum must approximately fit the normal distribution under a broad range of realistic conditions. The normal approximation is valid as long as the tails of the distribution are not too heavy, so that the variance were finite. Moreover, if the random summands have the moments of order higher than 2, then the normal approximation becomes more precise. The most interesting case is when the moment order lies between 2 and 3: the central limit theorem is still valid, but the random summands have so heavy tails that the third-order moment does not exist. Such heavytailed distributions are used, for example, for the analysis of the telecommunication system traffic. The present paper is devoted to the accuracy estimation of the normal approximation just in that case. We will present two-sided bounds for the constant in the Berry-Esseen inequality for Poisson random sums of independent identically distributed random variables with the finite moment order that lies between 2 and 3. The lower estimates obtained for the first time. We will improve the lower estimates and prove non-uniform estimates.

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“…where the supremum is taken over all distributions F of the random variable X 1 satisfying conditions (1). It can easily be seen that C(δ) ≥ C * (δ).…”

Section: The Lower Boundsmentioning

confidence: 99%

On the constants in the uniform and non-uniform versions of the Berry-Esseen inequality for Poisson random sums

Nefedova

Shevtsova

2010

International Congress on Ultra Modern Telecommunications and Control Systems

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“…Our reasoning is essentially based upon the techniques we developed in [2][3][4][5][6] for compound Poisson and compound Cox processes.…”

Section: N(t)mentioning

confidence: 99%

Asymptotic behavior of compound nonordinary Cox processes with nonzero means

Bening

Korolev

1998

J Math Sci

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“…The r.s. appears as models in many applied problems, for instance, in stochastic processes, stochastic modelling, random walk, queue theory, theory of network or theory of estimation, biology, nuclear physics, insurance, economic theory, finance mathematics and is essential in other fields too (see, e.g., [12][13][14][15]).…”

Section: Introductionmentioning

confidence: 99%

“…Some strong results for the asymptotic behaviour of the r.s., in case N has concrete probability laws: Poisson, Bernoulli, binomial or geometry, have been presented in the paper [16]. In 1997 Bening, Korolev and Shorgin considered three methods of the construction of approximations to the generalized Poisson distributions: approximation by a normal law, approximations by asymptotic distributions and approximation with the help of asymptotic expansions where uniform and non-uniform estimates are given (see [14]). Later, Korolev and Shevtsova (2012) presented sharpened upper bounds for the absolute constant in the Berry-Esseen inequality for Poisson and mixed Poisson r.s.…”

Section: Introductionmentioning

confidence: 99%

Large deviations for weighted random sums

Kasparavičiūtė

Saulis

2013

NAMC

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In the present paper we consider weighted random sums ZN = ∑j=1NajXj, where 0 ≤ aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throughout this paper N is independent of {X, Xj , j = 1, 2,...} and, for definiteness, it is assumed Z0 = 0. The main idea of the paper is to present results on theorems of large deviations both in the Cramér and power Linnik zones for a sum ~ZN = (ZN − EZN )(DZN )−1/2 , exponential inequalities for a tail probability P(~ZN > x) in two cases: µ = 0 and µ ≠ 0 pointing out the difference between them. Only normal approximation is considered. It should be noted that large deviations when µ ≠ 0 have been already considered in our papers [1,2].

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On approximations to generalized Poisson distributions

Cited by 20 publications

References 7 publications

On the constants in the uniform and non-uniform versions of the Berry-Esseen inequality for Poisson random sums

On the constants in the uniform and non-uniform versions of the Berry-Esseen inequality for Poisson random sums

Asymptotic behavior of compound nonordinary Cox processes with nonzero means

Large deviations for weighted random sums

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