Probabilities of Large Deviations on the Whole Axis

Rozovskii, Leonid Victorovich

doi:10.1137/1138005

Cited by 43 publications

(64 citation statements)

References 7 publications

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“…A nice review of recent developments in precise large deviations is given in Mikosch and Nagaev (1998). Other reviews can be found in Nagaev (1979) and Rozovski (1993). See also the monographs Vinogradov (1994), Gnedenko and Korolev (1996) and Meerschaert and Scheffler (2001).…”

Section: Introductionmentioning

confidence: 99%

Precise large deviations for sums of random variables with consistently varying tails

Tang

Yan

et al. 2004

Journal of Applied Probability

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Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation µ > 0. Under the assumption that the tail probability F (x) = 1 − F (x) is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N (t) , where N (t) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

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Section: Introductionmentioning

confidence: 99%

Precise large deviations for sums of random variables with consistently varying tails

Tang

Yan

et al. 2004

Journal of Applied Probability

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“…For instance, from the results of [20] it follows that such asymptotics hold for regularly varying distributions. The results of [27] imply that the same holds for Weibull-type distributions with a parameter smaller than 1/2. In [9] it is shown that Theorem 2.1 includes all the results known beforehand.…”

Section: Heavy-tailed Distributions Imentioning

confidence: 54%

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Denisov

Shneer²

2013

Journal of Applied Probability

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We study the exact asymptotics for the distribution of the first time τ x a Lévy process X t crosses a negative level −x. We prove that P(τ x > t) ∼ V (x)P(X t ≥ 0)/t as t → ∞ for a certain function V (x). Using known results for the large deviations of random walks we obtain asymptotics for P(τ x > t) explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.

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“…Remark 6. In the statement of [27, theorem 5a] it is not said that y * ≤ t − √ n, but one can find this assertion in the proof of [27,Lemma 3a].…”

Section: Theorem 23 ([27 Theorem 5a]) Let P{ξ 1 > Y} ∼ E −G(y) As mentioning

confidence: 99%

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Denisov

Shneer²

2013

J. Appl. Probab.

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Probabilities of Large Deviations on the Whole Axis

Cited by 43 publications

References 7 publications

Precise large deviations for sums of random variables with consistently varying tails

Precise large deviations for sums of random variables with consistently varying tails

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

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