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

Sotirios Losidis (), Konstadinos Politis () and Georgios Psarrakos ()
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Sotirios Losidis: University of Piraeus
Konstadinos Politis: University of Piraeus
Georgios Psarrakos: University of Piraeus

Methodology and Computing in Applied Probability, 2021, vol. 23, issue 4, 1489-1505

Abstract: Abstract The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

Keywords: Renewal process; Renewal density; Forward recurrence time; Backward recurrence time; Spread; Joint moments; Failure rate; Upper orthant stochastic order; 60K05; 60K10 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s11009-020-09787-w

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