Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

Losidis, Sotirios; Politis, Konstadinos; Psarrakos, Georgios

doi:10.1007/s11009-020-09787-w

Cited by 6 publications

(3 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. [15] prove that the survival function of the forward and backward recurrence time is given by…”

Section: Proposition 2 If the Distribution F Of The Interarrival Times Is Dfr Thenmentioning

confidence: 98%

“…When (γ t , δ t ) ≥ uo (γ s , δ s ), we can prove that F is NBUC by reversing the inequalities above. [15] proved that, under the assumption of DFR interarrival times, it holds that P(γ t > x) ≥ P(δ t > x), (11) for 0 ≤ x ≤ t.…”

Section: Propositionmentioning

confidence: 99%

“…Proof. [15] proved that under the assumption of DFR interarrival times E(γ t ) ≥ E(δ t ). Gakis and Sivazlian (1994) [9] proved that lim t→∞ E(γ t ) = E(γ ∞ ) = μ 2 /(2μ) = E(δ ∞ ).…”

Section: By Proposition 3 It Is Clear Thatmentioning

confidence: 99%

See 2 more Smart Citations

Covariance between the forward recurrence time and the number of renewals

Losidis¹

2021

Modern Stochastics: Theory and Applications

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…Proof. [15] prove that the survival function of the forward and backward recurrence time is given by…”

Section: Proposition 2 If the Distribution F Of The Interarrival Times Is Dfr Thenmentioning

confidence: 98%

Section: Propositionmentioning

confidence: 99%