Moments of the Forward Recurrence Time in a Renewal Process

Losidis, Sotirios; Politis, Konstadinos

doi:10.1007/s11009-018-9681-9

Cited by 6 publications

(4 citation statements)

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“…Losidis and Politis (2020) [14] study the covariance between the forward recurrence time and the number of renewals up to time t in an ordinary renewal process when t → ∞, and link the sign of the asymptotic covariance with the equilibrium distribution F e . However, it would be preferable to link the sign with the distribution F of the interarrival times instead of the F e .…”

Section: The Asymptotic Covariance Of the Forward Recurrence Time And The Number Of Renewalsmentioning

confidence: 99%

“…In literature, it seems that so far little attention has been given to this topic. Exceptions are Coleman (1982) [5] and Losidis and Politis (2020) [14]. More specifically, Coleman (1982) [5] proves that the covariance of the forward recurrence time and the number of renewals is given by…”

Section: Introductionmentioning

confidence: 99%

“…In literature, there are several sources which study the forward recurrence time, see, for example, Brown (1980) [2] and (1981) [3], Shaked and Zhu (1992) [17], Gakis and Sivazlian (1992) [8] and (1994) [9] and most recently Losidis and Politis (2020) [14].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, we study the covariance between γ t and N(t) for t → ∞. In contrast to Losidis and Politis (2020) [14], we focus on the distribution function F of the interarrival times instead of the equilibrium distribution function F e . The last section has some concluding remarks.…”

Section: Introductionmentioning

confidence: 99%

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Covariance between the forward recurrence time and the number of renewals

Losidis¹

2021

Modern Stochastics: Theory and Applications

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Recurrence times and the number of renewals in (0, t] are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in (0, t] at a fixed time point t and when t → ∞ is studied assuming that the interarrival distribution belongs to certain ageing classes.Keywords Renewal process, remaining term, forward recurrence time, number of renewals, covariance 2010 MSC 60K05, 60K10 ∞ 0 x k dF (x) for the k-th moment of F around zero; for simplicity, we write μ rather than μ 1 for the first moment. Also we assume that

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Section: The Asymptotic Covariance Of the Forward Recurrence Time And The Number Of Renewalsmentioning

confidence: 99%