О Неравномерных Оценках Скорости Сходимости В Центральной Предельной Теореме

Нефедова, Юлия Сергеевна; Nefedova, Yulia Sergeevna; Shevtsova, I. G.

doi:10.4213/tvp4432

Cited by 7 publications

(2 citation statements)

References 25 publications

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“…С другой стороны, с использованием результата работы [11] о том, что экстремум функционала, линейного по функции распределения случайной величины X, при одном линейном ограничении E X = a моментного типа достигается на некотором двухточечном распределении (также см. [2]), в статье [4] показано, что справедливо соотношение…”

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Уточнение Оценок Скорости Сходимости В Центральной Предельной Теореме При Отсутствии Моментов Порядков, Бо́льших Второго

Королев¹,

Korolev²,

Попов³

et al. 2011

Teor. Veroyatnost. i Primenen.

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Уточнены верхние оценки абсолютной постоянной в оценке скорости сходимости в центральной предельной теореме в терминах урезанных моментов. Показано, что абсолютная константа в неравенстве Осипова не превосходит 2.011. Ключевые слова и фразы: центральная предельная теорема, оценка скорости сходимости, абсолютная константа.

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“…Авторы выражают признательность Л. Падитцу за неоценимую библиографическую поддержку, а также Ю. С. Нефедовой и И. Г. Шевцовой за возможность ознакомиться с работой [4].…”

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Уточнение Оценок Скорости Сходимости В Центральной Предельной Теореме При Отсутствии Моментов Порядков, Бо́льших Второго

Королев¹,

Korolev²,

Попов³

et al. 2011

Teor. Veroyatnost. i Primenen.

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Nonuniform Approximations for Sums of Discrete m-Dependent Random Variables

Vellaisamy

Čekanavičius

2013

Contemporary Developments in Statistical Theory

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Non-uniform estimates are obtained for Poisson, compound Poisson, translated Poisson, negative binomial and binomial approximations to sums of of m-dependent integer-valued random variables. Estimates for Wasserstein metric also follow easily from our results. The results are then exemplified by the approximation of Poisson binomial distribution, 2-runs and m-dependent (k 1 , k 2 )-events.Nonuniform estimates for normal approximation are well known, see the classical results in Chapter 5 of [12] and the references [9], [10] and [19] for some recent developments. On the other hand, nonuniform estimates for discrete approximations are only a few. For example, the Poisson approximation to Poisson binomial distribution has been considered in [18] and translated Poisson approximation for independent lattice summands via the Stein method has been discussed in [2]. Some general estimates for independent summands under assumption of matching of pseudomoments were obtained in [6]. For possibly dependent Bernoulli variables, nonuniform estimates for Poisson approximation problems were discussed in [20]. However, the estimates obtained had a better accuracy than estimates in total variation only for x larger than exponent of the sum's mean. In [7], 2-runs statistic was approximated by compound Poisson distribution. In this paper, we obtain nonuniform estimates for Poisson, compound Poisson, translated Poisson, negative binomial and binomial approximations, under a quite general set of assumptions.We recall that the sequence of random variables {X k } k≥1 is called m-dependent if, for 1 < s < t < ∞, t − s > m, the sigma algebras generated by X 1 , . . . , X s and X t , X t+1 . . . are independent.Without loss of generality, we can reduce the sum of m-dependent variables to the sum of 1dependent ones, by grouping consecutive m summands. Therefore, we consider henceforth, without loss of generality, the sum S n = X 1 + X 2 + · · · + X n of non-identically distributed 1-dependent random variables concentrated on nonnegative integers.We denote the distribution function and the characteristic function of S n by F n (x) and F n (t), respectively. Similarly, for a signed measure M concentrated on the set N of nonnegative integers, we denote by M (x) = x k=0 M {k} and M (t) = ∞ k=0 e itk M {k}, the analogues of distribution function and Fourier-Stieltjes transform, respectively. Though our aim is to obtain the non-uniform estimates, we obtain also estimates for Wasserstein norm defined as M W = ∞ j=0 |M (j)|.Note that Wasserstein norm is stronger than total variation norm defined by M = ∞ j=0 |M {j}|.

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An upper bound for the absolute constant in the nonuniform version of the Berry-Esseen inequalities for nonidentically distributed summands

Григорьева

Popov

2012

Dokl. Math.

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О Неравномерных Оценках Скорости Сходимости В Центральной Предельной Теореме

Cited by 7 publications

References 25 publications

Уточнение Оценок Скорости Сходимости В Центральной Предельной Теореме При Отсутствии Моментов Порядков, Бо́льших Второго

Уточнение Оценок Скорости Сходимости В Центральной Предельной Теореме При Отсутствии Моментов Порядков, Бо́льших Второго

Nonuniform Approximations for Sums of Discrete m-Dependent Random Variables

An upper bound for the absolute constant in the nonuniform version of the Berry-Esseen inequalities for nonidentically distributed summands

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