A Note on the Closure of Convolution Power Mixtures (Random Sums) of Exponential Distributions

Albin, J.M.P.

doi:10.1017/s1446788708000104

Cited by 22 publications

(16 citation statements)

References 13 publications

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“…, it is enough to show that both H (1) and H (2) are long-tailed -see Property 2 (2). By the previous statement (2), we have H (1) ∈ L. Then the argument from [24] implies…”

Section: Finally the Last Term Is Not Bigger Thanmentioning

confidence: 99%

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Convolution and convolution-root properties of long-tailed distributions

Foss

Wang

2015

Extremes

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We obtain a number of new general properties, related to the closedness of the class of long-tailed distributions under convolutions, that are of interest themselves and may be applied in many models that deal with "plus" and/or "max" operations on heavy-tailed random variables. We analyse the closedness property under convolution roots for these distributions. Namely, we introduce two classes of heavytailed distributions that are not long-tailed and study their properties. These examples help to provide further insights and, in particular, to show that the properties to be both long-tailed and so-called "generalised subexponential" are not preserved under the convolution roots. This leads to a negative answer to a conjecture of Embrechts and Goldie [10,12] for the class of long-tailed and generalised subexponential distributions. In particular, our examples show that the following is possible: an infinitely divisible distribution belongs to both classes, while its Lévy measure is neither long-tailed nor generalised subexponential.

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“…, it is enough to show that both H (1) and H (2) are long-tailed -see Property 2 (2). By the previous statement (2), we have H (1) ∈ L. Then the argument from [24] implies…”

Section: Finally the Last Term Is Not Bigger Thanmentioning

confidence: 99%

“…✷ Proof of Theorem 2.2. Start with Proof of (1). Recall that F / ∈ L implies with necessity that F * n / ∈ S for all n ≥ 2.…”

Section: Finally the Last Term Is Not Bigger Thanmentioning

confidence: 99%

Convolution and convolution-root properties of long-tailed distributions

Foss

Wang

2015

Extremes

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show abstract

“…The random convolution closure for the class L was considered, for instance, in [1,14,16,17]. We now present a particular statement of Theorem 1.1 from [17].…”

Section: Introductionmentioning

confidence: 99%

Randomly stopped sums with consistently varying distributions

Kizinevič

Sprindys

Šiaulys

2016

Modern Stoch. Theory Appl.

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Let {ξ1, ξ2, . . .} be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for {ξ1, ξ2, . . .} and η under which the distribution function of the random sum Sη = ξ1 + ξ2 + · · · + ξη belongs to the class of consistently varying distributions. In our consideration, the random variables {ξ1, ξ2, . . .} are not necessarily identically distributed.

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“…The random closure results for class D can be found in [7,16], and for the class L, in [1,16,19,20]. The random closure results for the class C can be derived from the results of [14].…”

Section: Introductionmentioning

confidence: 99%

Randomly stopped maximum and maximum of sums with consistently varying distributions

Andrulytė

Manstavičius

Šiaulys

2017

Modern Stoch. Theory Appl.

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Let {ξ1, ξ2, . . .} be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let S0 := 0 and Sn := ξ1 + ξ2 + · · · + ξn for n 1. We consider conditions for random variables {ξ1, ξ2, . . .} and η under which the distribution functions of the random maximum ξ (η) := max{0, ξ1, ξ2, . . . , ξη} and of the random maximum of sums S (η) := max{S0, S1, S2, . . . , Sη} belong to the class of consistently varying distributions. In our consideration the random variables {ξ1, ξ2, . . .} are not necessarily identically distributed.

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A Note on the Closure of Convolution Power Mixtures (Random Sums) of Exponential Distributions

Abstract: We make a correction to an important result by Cline 2000 Mathematics subject classification: primary 60F99, 62E20; secondary 44A10, 60E07.

Cited by 22 publications

References 13 publications

Convolution and convolution-root properties of long-tailed distributions

Convolution and convolution-root properties of long-tailed distributions

Randomly stopped sums with consistently varying distributions

Randomly stopped maximum and maximum of sums with consistently varying distributions

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