The Minimal Growth Rate of Partial Maxima

Klass, Michael J.

doi:10.1214/aop/1176993296

Cited by 23 publications

(23 citation statements)

References 9 publications

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“…whereF = 1 − F and a n = F ← (1 − 1/n) (with F ← (t) = inf{x : F (x) ≥ t}). More recently, Klass [12] gave a different proof of the same result using a blocking technique argument. In this paper, the independence assumption is removed.…”

Section: Introductionmentioning

confidence: 94%

Almost sure relative stability of the maximum of a stationary sequence

Naveau¹

2003

Advances in Applied Probability

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The almost sure convergence of the maximum of a strictly stationary sequence is studied. We show that, if a particular mixing condition, related to the asymptotic independence of maxima defined by O'Brien, is satisfied, then the maximum behaves asymptotically like the quantile F←(1-1/n) whenever the marginal tail distribution decreases quickly enough. A necessary condition for the almost sure convergence of the maximum is derived. Applications are also discussed.

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Section: Introductionmentioning

confidence: 94%

Almost sure relative stability of the maximum of a stationary sequence

Naveau¹

2003

Advances in Applied Probability

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“…Proof of Lemma 2. Fix a sequence of standard exponential random variables (τ e i ) i∈N and assume without loss of generality that the sequence (z n ) n∈N is constructed from (τ e i ) i∈N via formula (16). The subsequent proof is divided into two steps.…”

Section: Preliminariesmentioning

confidence: 99%

The laws of iterated and triple logarithms for extreme values of regenerative processes

Marynych¹,

Matsak²

2020

Modern Stochastics: Theory and Applications

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We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the lim sup and a law of the triple logarithm for the lim inf. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424-445]. We apply our results to several queuing systems and a birth and death process.

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“…(iii) The monotonicity requirement on b t can be relaxed somewhat, analogous to the discrete situation, as discussed by Klass [11]. We omit the details.…”

Section: Lower Classes For (R) Atmentioning

confidence: 99%

“…(Resnick and Rubinovitch [17] mentioned continuous-time, small-, and largetime, versions of this result.) Klass [11], [12] removed the side condition, thus giving clean necessary and sufficient conditions for the partial maximum of a sequence of i.i.d. random variables to be in the lower class (according to our usage of the term).…”

Section: Further Background and Connectionsmentioning

confidence: 99%

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Small-time almost-sure behaviour of extremal processes

Maller

Schmidli

2017

Adv. Appl. Probab.

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An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

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The Minimal Growth Rate of Partial Maxima

Cited by 23 publications

References 9 publications

Almost sure relative stability of the maximum of a stationary sequence

Almost sure relative stability of the maximum of a stationary sequence

The laws of iterated and triple logarithms for extreme values of regenerative processes

Small-time almost-sure behaviour of extremal processes

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