Large deviations for distributions of sums of random variables: Markov chain method

Fatalov, V. R.

doi:10.1134/s0032946010020055

Cited by 6 publications

(3 citation statements)

References 23 publications

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“…Among the existing methods for large deviations (see, e. g., Saulis and Statulevičius 1991;Jensen 1995;Borovkov 1999;Fatalov 2011Fatalov , 2010 Zhao 2011), we rely on the cumulant method that was proposed by S. V. Statulevičius (1966) and developed by R. Rudzkis, L. Saulis, and V. Statulevičius (1978), as it is a powerful method that permits the systematic investigation of large deviations for various statistics.…”

Section: Applied Methodsmentioning

confidence: 99%

“…The papers (Aksomaitis 1965(Aksomaitis , 1967(Aksomaitis , 1973Statulevičius 1967;Saulis 1978Saulis , 1981Saulis and Deltuvienė 2007) address large deviations in the Cramér zone for distributions of random sums. Of existing methods on large deviations (see, e. g., Saulis and Statulevičius 1991;Jensen 1995;Borovkov 1999;Fatalov 2011Fatalov , 2010; Gao and Zhao 2011), we rely on the cumulant method. However, there are only a small number of papers (see, e. g., Statulevičius 1967;Saulis 1978Saulis , 1981Saulis and Deltuvienė 2007;Kasparavičiūtė and Saulis 2010, 2011a, 2011b, 2013 on normal approximation taking into account large deviations for the distribution of the sums of a r. n. s. in the case where cumulant method is used.…”

Section: Limit Theorems For Compound Sumsmentioning

confidence: 99%

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Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables

Kasparavičiūtė¹

2013

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In probability theory, the topic of large deviations, i. e., approximation problems of the probabilities of rare events, have a significant place. To understand why rare events are important at all one only has to think of the events in an insurance mathematics, nuclear physics and etc., to be convinced that those events can have an enormous impact. This thesis is concerned with a normal approximation to a distribution of the sum Z N = N j=1 a j X j , Z 0 = 0, 0 < a j < ∞, of a random number of summands N of independent identically distributed weighted random variables {X, X j , j = 1, 2, ...} that takes into consideration large deviations in both the Cramér zone (the characteristic functions of the summands of Z N are analytic in a vicinity of zero) and the power Linnik zone (the growth of the moments of the summands does not ensure the analyticity of the characteristic functions). Here a non-negative integer-valued random variable N is independent of {X, X j , j = 1, 2, ...}. In addition, the asymptotic expansion that take into consideration large deviations in the Cramér zone for the density function of the standardized compound Poisson process is obtained. To solve the problems, the classical method of characteristic functions, cumulant and combinatorial methods are used. Although, in probability theory the asymptotic behavior of tail probabilities for the sums of a random number of summands of random variables is a quite new problem, it was initiated in the XXth century, but there is a very extensive literature on mentioned problem. However, as it is known for the author of the dissertation, there are a few scientific works on theorems of large deviations for the sums of a random number of summands of independent random variables in case where the cumulant method is used. The thesis consists of an introduction, three chapters, general conclusions, references, and a list of the author's publications. The introduction reveals the importance of the scientific problem, describes the tasks of the thesis, research methodology, scientific novelty, the practical significance of results. In the first chapter an overview of the problems is presented. The second chapter is devoted for obtaining an upper bound for the cumulants, theorems of large deviations and exponential inequalities for the standardized version of the sum Z N. The instances of large deviations (the law of N is known; a j ≡ 1; discount version of large deviations) are also analyzed in this chapter. In the third chapter, the asymptotic expansion of large deviations in the Cramér zone for the density function of the standardized compound Poisson process is considered. i. i. d.-independent identically distributed; r. n. s.-random number of summands; a. s.-almost sure.

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Section: Applied Methodsmentioning

confidence: 99%

Section: Limit Theorems For Compound Sumsmentioning

confidence: 99%

Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables

Kasparavičiūtė¹

2013

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“…Имеющиеся у автора доказательства следствий 1.1, 1.2 довольно длинны и в настоящей статье не приводятся. Нетривиальный аналог теорем 1.1, 1.2 для простейшей марковской цепи, представляющей собой последовательность независимых одинаково распределенных случайных величин, доказан в [34]. § 2.…”

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Точные Асимптотики Вероятностей Больших Уклонений Для Цепей Маркова: Метод Лапласа

Fatalov¹

2011

Izv. RAN. Ser. Mat.

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Доказаны результаты о точных асимптотиках при n → ∞ для средних Ea exp −θ n−1 k=0 g(X k) и вероятностей Pa 1 n n−1 k=0 g(X k) < d , где Xn = X0 + n k=1 ξ k , n 1,-соответствующее случайное блуждание на R, {ξ k } ∞ k=1-последовательность независимых одинаково распределенных случайных величин, имеющих распределение Лапласа, g(x)-непрерывная положительная функция, удовлетворяющая некоторым условиям, и d > 0, θ > 0, a ∈ R-заданные числа. Результаты получены на основе развитого в статье нового метода-метода Лапласа для времени пребывания марковских цепей с дискретным временем. В качестве функции g(x) можно брать функции |x| p , log(|x| p + 1), p > 0, |x| log(|x| + 1), e α|x| − 1, 0 < α < 1/2, x ∈ R, и другие. Подробно рассмотрен пример с функцией g(x) = |x|, в котором проведены явные вычисления с привлечением функций Бесселя. Библиография: 44 наименования.

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Asymptotic Expansion for the Distribution Density Function of the Compound Poisson Process in Large Deviations

Kasparavičiūtė

Deltuvienė

2016

J Theor Probab

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Large deviations for distributions of sums of random variables: Markov chain method

Cited by 6 publications

References 23 publications

Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables

Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables

Точные Асимптотики Вероятностей Больших Уклонений Для Цепей Маркова: Метод Лапласа

Asymptotic Expansion for the Distribution Density Function of the Compound Poisson Process in Large Deviations

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