Modern Stochastics: Theory and Applications

A renewal process considers the sequence of partial sums $\{{S_{n}}:n=0,1,2,\dots \}$ of some independent, nonnegative random variables ${X_{i}}$, so that ${S_{0}}=0$ and, for $n\ge 1$, ${S_{n}}={X_{1}}+{X_{2}}+\cdots +{X_{n}}$. We also assume that ${X_{i}}$ ($i=1,2,\dots $) has a distribution function (d.f.) F. We assume throughout the paper that F is absolutely continuous and has density f. We write ${\mu _{k}}={\textstyle\int _{0}^{\infty }}{x^{k}}dF(x)$ for the k-th moment of F around zero; for simplicity, we write μ rather than ${\mu _{1}}$ for the first moment. Also we assume that F has a finite variance ${\sigma ^{2}}$. Further, the equilibrium distribution associated with F, denoted by ${F_{e}}$, is defined as ${F_{e}}(t)={\mu ^{-1}}{\textstyle\int _{0}^{t}}\overline{F}(y)dy$, where $\overline{F}=1-F$.

A renewal process is the counting process $\{N(t):t\ge 0\}$ whereby $N(t)=\sup \{n:{S_{n}}\le t\}$. $N(t)$ is the number of renewals that occurred in $(0,t]$ as defined above.

However, the distribution of $N(t)$ is not always easy to calculate, which makes the renewal function $U(t)=E[N(t)]={\textstyle\sum _{n=1}^{\infty }}\mathbb{P}({S_{n}}\le t)$, for $t\ge 0$, a quantity of interest. Assuming that F is absolutely continuous, the renewal function U is also absolutely continuous with a density u, the renewal density. We mention for future reference that a version of u satisfies the renewal equation see, for example, Feller (1971, Ch. XI) [7].

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].

We define by ${\gamma _{t}}={S_{N(t)+1}}-t$ the forward recurrence time and by ${\delta _{t}}=t-{S_{N(t)}}$ the backward recurrence time.

Gakis and Sivazlian (1992) [8] compute the joint distribution of the backward and forward recurrence times using the joint distribution $\mathbb{P}({\gamma _{t}}\le y,{\delta _{t}}\le x,N(t)=n>0)$, namely The above relation triggered this paper’s main topic: to investigate the relationship that governs recurrence times and $N(t)$. 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 He also proves that when $t\to \infty $ Losidis and Politis (2020) [14], using the remaining term $Q(t)$ defined as prove that the asymptotic covariance between the forward recurrence time and the number of renewals on $(0,t)$ is given by Assuming that $U(t)$ is absolutely continuous, $Q(t)$ is also continuous with ${Q^{\prime }}(t)=q(t)$.

We define the failure rate by and we say that the distribution F has decreasing (increasing) failure rate (denoted by DFR (IFR)) if the quantity $\overline{F}(t-x)/\overline{F}(t)$ is decreasing (increasing) in t ($t>0$) for all $x\ge 0$.

A wider class compared to DFR (IFR) is the IMRL (DMRL). We define by $m(t)=\mu {\overline{F}_{e}}(t)/\overline{F}(t)$ the mean residual life. We say that F has increasing (decreasing) mean residual life (we denote IMRL (DMRL) respectively), if $m(t)$ is increasing (decreasing) function. For more details, see Willmot and Lin (2001) [18]. Another class is the new worse (better) than used in convex ordering, named as the NWUC (NBUC) class. We say F is NWUC (NBUC) if In this paper, we tackle the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$, a topic that escaped attention so far.

The main novelty of this paper is that under the assumption of IMRL interarrival times, we prove that the covariance between the forward recurrence time and the number of renewals in $(0,t]$ is less or equal to zero.

We organise the remaining of this paper as follows.

In Section 2, initially, we discuss the upper orthant stochastic order for the family of random pairs $({\gamma _{t}},{\delta _{t}})$ and the hazard rate order between the backward and the forward recurrence times in a renewal process. Next, we focus on the covariance between the forward recurrence time and the number of renewals in $(0,t]$ for $t<\infty $.

In Section 3, we study the covariance between ${\gamma _{t}}$ and $N(t)$ for $t\to \infty $. 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.

Recurrence times find numerous real-life applications. In addition to traditional application areas, such as reliability or actuarial risk theory, renewal processes have been used for several decades in biostatistics, in models for the early detection of disease (see, e.g., Zelen and Lee (2002) [19]).

To give a more specific example, from medicine, we can define the forward recurrence time as the time until a person leaves the disease state after detection at time t (in other words, the forward recurrence time represents the time that a patient knows he has a condition until recovery). The backward recurrence time would be the time from when the person has had the disease until the diagnosis.

We use this example to illustrate the critical question: given the time passed since the person was ill (this could be considered as the latest renewal), how long until they get infected again from the same decease (this shall be the next renewal)? Mathematically, this is best formulated by considering the joint distribution of the (backward and forward) recurrence times associated with a renewal process.

It is known (see, for example, Daley and Vere-Jones (2003) [6]) that for any $0\le x\le t$ and $y\ge 0$ it holds that In a multivariate setting, there are various ways to generalise the concept of the usual, univariate stochastic ordering. One of them is the upper orthant ordering, as discussed, for example, in Shaked and Shanthikumar (2007, Ch. 6) [16]. More explicitly, let $\mathbf{X}=({X_{1}},{X_{2}},\dots ,{X_{n}})$ and $\mathbf{Y}=({Y_{1}},{Y_{2}},\dots ,{Y_{n}})$ be two n-dimensional random vectors. Then we say that X is smaller that Y in the upper orthant stochastic order, and we denote this by $\mathbf{X}{\le ^{uo}}\mathbf{Y}$ if for any ${t_{1}},{t_{2}},\dots ,{t_{n}}$, Li et al. (2000) [11] showed that if ${\gamma _{t}}$ is increasing in $t\ge 0$ in convex ordering then F is NWUC. Next we present a result similar to that of Li et al. (2000) [11], for the family of random pairs $({\gamma _{t}},{\delta _{t}})$.

Losidis et al. (2020) [15] proved that, under the assumption of DFR interarrival times, it holds that for $0\le x\le t$.

Recall that for two random variables X and Y supported on $[0,\infty )$ with survival functions $P(X>x)$ and $P(Y>y)$, we say that X is less than or equal to Y in the usual stochastic order ($X{\le _{st}}Y$) if $P(X>x)\le P(Y>y)$ for all $t\ge 0$ (see Shaked and Shanthikumar (2007, Ch. 1)) [16]. In the terminology of stochastic orders, (11) means that ${\delta _{t}}$ is smaller than ${\gamma _{t}}$ in the usual stochastic order. The next result, which in view of the last equation is an improvment of this, shows that ${\delta _{t}}$ is smaller that ${\gamma _{t}}$ in the hazard rate order (for details on those, see Shaked and Shanthikumar (2007) [16]).

Next, we present a relation for the conditional tail functions of the forward and backward recurrence times.

Continuing the example we began in this section, one could enquire the following: is there a connection between the number of occasions we may get sick from a virus and the period it takes to recover since we tested positive to it? Mathematically this could be approached by the covariance between the forward recurrence time and the number of renewals.

In the following result, initially, we provide an alternative formula for the $Cov({\gamma _{t}},N(t))$ in terms of the remaining term $Q(t)$, which enables us to show that, assuming IMRL interarrival times, $Cov({\gamma _{t}},N(t))\le 0$.

In the next result we study the difference $Cov({\gamma _{t}},N(t)-{\lim \nolimits_{t\to \infty }}Cov({\gamma _{t}},N(t))$. More specifically, we define If $S(t)\ge (\le )0$, then the limit of $Cov({\gamma _{t}},N(t))$ as $t\to \infty $ serves as a bound for the $Cov({\gamma _{t}},N(t))$. In the sequel we present conditions under which the asymptotic covariance between the forward recurrence time and the number of renewals can be used as an upper bound for $Cov({\gamma _{t}},N(t))$. We begin with a formula which calculates the difference $Cov({\gamma _{t}},N(t))-{\lim \nolimits_{t\to \infty }}Cov({\gamma _{t}},N(t))$.

Conversely to the IMRL assumption of interarrival times ($Q(t)\ge 0$ and $q(t)\le 0$), we next study the case when $Q(t)$ is less than or equal to zero and also is an increasing function. In this case we prove that the asymptotic covariance between the forward recurrence time and the number of renewals can be used as an upper bound for the $Cov({\gamma _{t}},N(t))$.

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\to \infty $, 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 ${F_{e}}$.

Next, we use stochastic order results between ${\gamma _{t}}$ and ${\gamma _{\infty }}$ in order to determinate the sign of ${\lim \nolimits_{t\to \infty }}Cov({\gamma _{t}},N(t))$.

More precisely, as $t\to \infty $, the random variable ${\gamma _{t}}$ converges in distribution to the variable ${\gamma _{\infty }}$, with ${\lim \nolimits_{t\to \infty }}\mathbb{E}({\gamma _{t}})=\mathbb{E}({\gamma _{\infty }})={\mu _{2}}/(2\mu )$. The next result presents a formula for ${\lim \nolimits_{t\to \infty }}Cov({\gamma _{t}},N(t))$ in terms of the variables ${\gamma _{t}}$ and ${\gamma _{\infty }}$.

Shaked and Zhu (1992) [17] proved that the renewal function is convex (concave) if, and only if, the excess lifetime ${\gamma _{t}}$ is stochastically decreasing (increasing). They also proved that if the renewal function is convex (concave), then the distribution F of the interarrival time is NBUE (NWUE). Combining this with Eq. (33) and under the assumption that renewal function $U(t)$ is convex (concave), then we get that the distribution F of the interarrival times is NBUE (NWUE) and the asymptotic covariance between the forward recurrence time and the number of renewals is more (less) or equal to zero.

By Proposition 3 it is clear that, if $\mathbb{E}({\gamma _{\infty }})\ge (\le )(\mathbb{E}({\gamma _{t}})$ then However, it is known (see, e.g., Brown (1980) [2] or Shaked and Zhu (1992) [17]) that, in contrast to the IMRL case, the condition that $m(t)$ is a decreasing function does not guarantee that $\mathbb{E}({\gamma _{t}})$ is decreasing, and Corollary 3 does not hold for the reverse inequality, i.e. assuming DMRL interarrival times. However, it can be proven considering IFR interarrival times. More specifically, the following theorem holds.

For the remaining of this section, we focus on the asymptotic covariance between the variables ${\gamma _{t}^{r}}$ (for $r=1,2,\dots $) and $N(t)$, namely ${\lim \nolimits_{t\to \infty }}Cov({\gamma _{t}^{r}},N(t))$.

Gakis and Sivazlian (1994) [9] study the asymptotic covariance between the forward and backward recurrence times, denoted as Combining (2) and (34) yields that Substituting ${\gamma _{t}}$ with ${\gamma _{t}^{r}}$ in Eq. (35) gives Next, we show that this quantity plays a key role for known bounds for the renewal function $U(t)$ under the assumption of IMRL interarrival times.

Brown (1980) [2], assuming that the distribution function F of the interarrival times is IMRL, gives the following lower bound for the renewal function: for $r=1,2,\dots ,k$, with where

Next, we present an alternative formula for the ${\lim \nolimits_{t\to \infty }}Cov({\gamma _{t}^{r}},N(t))$. More specifically, the following proposition holds.

The following corollary is immediate.