On upper bounds for the tail distribution of geometric sums of subexponential random variables

Richards, Andrew

doi:10.1007/s11134-009-9125-x

Cited by 4 publications

(8 citation statements)

References 21 publications

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“…In the case when (1) is not fulfilled, upper bounds for P(M > x) have been derived by Kalashnikov [11] and by Richards [17]. The approach in these papers is based on the representation of M as a geometric sum of independent random variables:…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Upper Bounds for the Maximum of a Random Walk with Negative Drift

Kugler

Wachtel

2013

Journal of Applied Probability

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Abstract. Consider a random walk Sn = n i=0 X i with negative drift. This paper deals with upper bounds for the maximum M = max n≥1 Sn of this random walk in different settings of power moment existences. As it is usual for deriving upper bounds, we truncate summands. Therefore we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for Mτ = max n≤τ Sn. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…This even remains valid, if we bound E[τ z ] in these inequalities by combining (15) or (16) with (14). The reason why we are able to obtain asymptotically precise bounds is, that we may choose z arbitrary large.…”

Section: Theorem 1 Assume Thatmentioning

confidence: 93%

Upper Bounds for the Maximum of a Random Walk with Negative Drift

Kugler

Wachtel

2013

Journal of Applied Probability

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“…Comparing this with (4), we see that the inequalities in Theorem 7 are asymptotically precise. This even remains valid, if we bound E[τ z ] in these inequalities by combining (15) or (16) with (14). The reason why we are able to obtain asymptotically precise bounds is, that we may choose z arbitrary large.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 85%

Upper Bounds for the Maximum of a Random Walk with Negative Drift

Kugler¹,

Wachtel²

2013

J. Appl. Probab.

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Consider a random walk S n = n i=0 X i with negative drift. This paper deals with upper bounds for the maximum M = max n≥1 S n of this random walk in different settings of power moment existences. As is usual for deriving upper bounds, we truncate summands. Therefore, we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby, we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for M τ = max n≤τ S n . In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

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“…Nevertheless some authors (see, e.g. [1,16]) show that the relative error in (1) can be big even for rather large values of x. The second way is to derive explicit bounds for the difference of the left and right sides of (1).…”

Section: Introductionmentioning

confidence: 99%

“…Such an approach was developed for regular varying and Weibull distributions B in [7][8][9][10]16]. For general τ, this approach is even less effective because of necessity for explicit estimates for renewal measure generated by the descending ladder hight.…”

Section: Introductionmentioning

confidence: 99%