Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process
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Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process Sotirios Losidis1 · Konstadinos Politis1 · Georgios Psarrakos1 Received: 4 April 2019 / Revised: 6 November 2019 / Accepted: 25 March 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 Mathematics Subject Classification (2010) 60K05 · 60K10
1 Introduction Let X1 , X2 , . . . , be a sequence of independent, identically distributed (i.i.d.) nonnegative random variables with distribution function (d.f.) F . Define S0 = 0 and, for n = 1, 2, . . . , let Sn = X1 + X2 + · · · + Xn . A renewal process is the counting process {N (t) : t ≥ 0}, where N (t) = sup{n : Sn ≤ t}, so that N (t) represents the number of partial sums (renewal Konstadinos Politis
[email protected] Sotirios Losidis losidis [email protected] Georgios Psarrakos [email protected] 1
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Demetriou Str, Piraeus 18534, Greece
Methodology and Computing in Applied Probability
epochs) Sn which are less than or equal to t. A key quantity of interest is the renewal function U (t) = E[N (t)], and it is well-known that U (t) = ∞ n=1 P(Sn ≤ t), for t ≥ 0. A question which arises naturally and is important in various contexts is the following: given the time that has elapsed since the last renewal, how long one has to wait until the next renewal occurs? Mathematically, this is best formulated by considering the joint distribution of the (backward and forward) recurrence times associated with a renewal process, which we now define. Let t ≥ 0. The backward recurrence time, or the age at time t, denoted by βt and defined by βt = t − SN(t) , is the time that has passed since the last renewal occurred. The forward recurrence time at time t (also known as the excess lifetime), denoted by γt , is defined by γt = SN(t)+1 − t and it is clear that it represents the length of time from t until the occurrence of the next renewal. We shall assume throughout the paper that the distribution of interarrival times (or the interrenewal distribution) F is absolutely continuous with a density f . Moreover, in Sections 3 and 4 we sha
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