Modern Stochastics: Theory and Applications

Two of the quantities of primary interest in renewal theory are the forward and backward recurrence times, denoted by ${\gamma _{t}}$ and ${\delta _{t}}$, respectively. On one hand, their asymptotic distribution is intimately related to the renewal theorem; see, for instance, Resnick [19] and Omey and Teugels [17]. On the other hand, and from a more practical viewpoint, if at time t in a renewal process one is interested in the probability that there will be no renewals for y time units ahead, this is simply $\mathbb{P}({\gamma _{t}}\gt y)$. Moreover, if we know the current age of the item in operation at time t (that is, the backward recurrence time) one expects that, in ‘most cases,’ we have a better view of the chance that there will be a gap with no renewals for y time units from now. For, if the current age is x, the required probability is now $\mathbb{P}({\gamma _{t}}\gt y\hspace{0.1667em}|\hspace{0.1667em}{\delta _{t}}=x)$.

To be more precise, it is well-known that the two recurrence times are independent if and only if the underlying stochastic model is that of a Poisson process; for the more general case of a renewal process, one expects that the dependence structure of the pair $({\gamma _{t}},{\delta _{t}})$ reflects in some way the aging pattern of the interarrival distribution F. Our first aim in the present paper is to explore this link and study conditions under which a certain dependence form of the pair $({\gamma _{t}},{\delta _{t}})$ implies an aging property for F and vice versa. Moreover, aging properties for F are well-known to be related to the concavity of the renewal function and the stochastic monotonicity of ${\gamma _{t}}$ and ${\delta _{t}}$; see Brown [4] and Shaked and Zhu [21]. Note that, when we say, e.g., that the forward recurrence times are stochastically increasing, this means that ${\gamma _{t}}$ is smaller, in the usual univariate stochastic order, than ${\gamma _{s}}$ for any $0\le t\lt s$; that is, for any $y\ge 0$, it holds that $\mathbb{P}({\gamma _{t}}\gt y)\le \mathbb{P}({\gamma _{s}}\gt y)$.

In Section 4 of the present paper we obtain results that link the stochastic monotonicity of the pair $({\gamma _{t}},{\delta _{t}})$ with aging properties for the interarrival distribution F and the concavity of the renewal function. Further, in Section 5 we discuss how similar properties of the conditional tail for one of the recurrence times, given the other, provide information for F and vice versa. Our results extend and generalize those obtained earlier by Brown [4], Chen [5] and Losidis et al. [15]. Further, aging properties and monotonicity results for the forward recurrence times were studied by Polatioglu and Sahin [18]; for diverse areas of potential application of the recurrence times in practice, see Losidis [12, 13].

To begin with, a renewal counting process $\{N(t):t\ge 0\}$ is a stochastic process that registers the successive occurrences of an event through time. We assume that the time between the $(i-1)$th and the ith such occurrence (interarrival time) is denoted by a random variable ${X_{i}}$ and that ${X_{1}},{X_{2}},\dots $ is a sequence of independent and identically distributed (i.i.d.) nonnegative random variables with distribution function F. For each t, the variable $N(t)$ then represents the number of events until time t; note that, in contrast to some books in this area, we do not assume the occurrence of an event at time zero. If we set ${S_{0}}=0$ and for $n=1,2,\dots $, ${S_{n}}={X_{1}}+{X_{2}}+\cdots +{X_{n}}$, the recurrence times are defined as follows: the forward recurrence time ${\gamma _{t}}={S_{N(t)+1}}-t$ denotes the time, from time t, until the next event (renewal), while ${\delta _{t}}=t-{S_{N(t)}}$ is the time elapsed since the last renewal. Another quantity of key interest is the renewal function $U(t)=E[N(t)]$. We assume throughout that the interarrival distribution F is absolutely continuous with a density f, so that U also has a density which is denoted by $u(t)$, and is called the renewal density.

For a distribution function F, we write $\overline{F}=1-F$ for the associated tail (or survival) function. We shall use the following well-known aging classes of distributions on the nonnegative half-line. A distribution F is said to have an increasing failure rate (IFR) if $\log \overline{F}$ is concave, that is, for all $x\gt 0$ the function $\overline{F}(x+t)/\overline{F}(t)$ is decreasing in t for all t such that $\overline{F}(t)\gt 0$. If F has a density f, this is equivalent to the condition that $f(t)/\overline{F}(t)$ is increasing in t, for t such that $\overline{F}(t)\gt 0$. Similarly, if $\log \overline{F}$ is convex, then F is called a decreasing failure rate (DFR) distribution. Next, a distribution F is called new better than used (NBU) if $\overline{F}(x+t)\le \overline{F}(x)\hspace{0.1667em}\overline{F}(t)$ for all $x,t\ge 0$, while F is called new worse than used (NWU) if $\overline{F}(x+t)\ge \overline{F}(x)\hspace{0.1667em}\overline{F}(t)$ for all $x,t\ge 0$. It is known that the DFR (IFR) is a subclass of the NWU (resp., NBU) class of distributions. Further details on these and other aging classes of distributions can be found in the books of Shaked and Shanthikumar [20] and Willmot and Lin [22].

For a nonnegative random variable X, which in our context typically represents lifetime, the variable ${X_{t}}=X-t\hspace{0.1667em}|\hspace{0.1667em}X\gt t$ denotes the residual lifetime at age t. We write ${F_{t}}$ for the distribution of ${X_{t}}$.

Further, for a distribution function F with finite first moment μ, the associated equilibrium (or stationary renewal) distribution, denoted by ${F_{e}}$ is defined for $x\ge 0$ by ${F_{e}}(x)={\mu ^{-1}}\hspace{0.1667em}\hspace{-0.1667em}{\textstyle\int _{0}^{x}}\overline{F}(y)dy$. Finally, we note that throughout the paper the term increasing (decreasing) stands for nondecreasing (nonincreasing).

The outline of the paper is as follows: in the next section we recall the main types of dependence for pairs of random variables. In Section 3, we discuss how a dependence structure for the pair $({\gamma _{t}},{\delta _{t}})$ is linked with an aging property for the interarrival distribution F. Next, our results in Sections 4 and 5 demonstrate the interplay between an aging pattern for F and various monotonicity properties for the joint and the conditional tail of the distribution for the pair $({\gamma _{t}},{\delta _{t}})$. We also show that the renewal function is concave if and only if the pair $({\gamma _{t}},{\delta _{t}})$ is increasing with respect to the upper orthant stochastic order. Our results are illustrated by two numerical examples in Section 6.

In this section we recall some well-known dependence concepts and stochastic orders of two-dimensional random variables that will be needed in the sequel. For further details the reader is referred to the books by Gupta et al. [9] and Denuit et al. [6].

The pair $(X,Y)$ of nonnegative random variables is called For future reference, we mention that if the distribution of X has finite support, then (3) is only required to hold for v, z in the support of ${F_{X}}$. Further, we note that in the bivariate case that we consider here, the condition in (1) is equivalent to the condition that for $x,y\ge 0$, The dual concepts of dependence are defined by reversing the sign of the inequalities above. For example, X, Y are negatively quadrant dependent (NQD) if for any $x,y\ge 0$, Another related notion is that of association, introduced by Esary et al. [7]. We say that the random variables X and Y are (positively) associated if for any bivariate increasing (in both dimensions) functions $g(x,y)$ and $h(x,y)$, provided of course that all the relevant expectations exist. If the sign of the above inequality is reversed, we say that the variables are negatively associated.

Among the various types of dependence mentioned above, positive regression dependence is the strongest, as every other dependence structure that we have defined is implied by it. In particular, we have the following implications [9]: It is important to note in our context that, as pointed out by Lehmann [11] and in contrast, e.g., to PQD or association, PRD is not symmetric in X and Y; that is, the fact that the pair $(X,Y)$ is PRD does not necessarily imply that the same is true for $(Y,X)$. However, it is argued in Lehmann [11] that if one of these two pairs is PRD, this suffices to yield the other types of dependence for the variables X and Y.

In the following sections we shall compare the pairs $({\gamma _{t}},{\delta _{t}})$ for different values of t. This leads us to the notion of a bivariate stochastic order. Univariate stochastic comparisons of the excess lifetime at different times of a renewal process have been studied by Belzunce et al. [2]. In fact, there are various ways to generalize the usual (univariate) stochastic ordering between two random variables X and Y. A standard reference for multivariate stochastic orders is Shaked and Shanthikumar [20, Chap. 6]; see also Belzunce et al. [1, Chap. 3]. Here, we shall confine ourselves to the notion of upper orthant stochastic order for two pairs of random variables.

More precisely, let $({X_{1}},{X_{2}})$ and $({Y_{1}},{Y_{2}})$ be two pairs of random variables. Then we say that $({X_{1}},{X_{2}})$ is smaller than $({Y_{1}},{Y_{2}})$ in the upper orthant order (denoted by $({X_{1}},{X_{2}}){\le _{\mathit{uo}}}({Y_{1}},{Y_{2}})$) if for any x, y, It is clear that the upper orthant order implies the univariate stochastic order of the component variables, so that from the last expression we obtain $\mathbb{P}({X_{1}}\gt x)\le \mathbb{P}({Y_{1}}\gt x)$ and $\mathbb{P}({X_{2}}\gt y)\le \mathbb{P}({Y_{2}}\gt y)$ for any $x,y\ge 0$.

Here we study the dependence structure of the recurrence times ${\gamma _{t}}$, ${\delta _{t}}$ in a renewal process at the same time point t. More explicitly, let X denote an interarrival time in the process, having distribution function F, and let $({\gamma _{t}},{\delta _{t}})$ be the pair of the forward and backward recurrence times, respectively, at time t. Then we have the following result. Here, NRD stands for negatively regression dependent, the dual dependence relation of PRD.

Because of the last proposition, and bearing in mind the discussion in the previous section, we see that if F has a monotone failure rate, then the pair $({\gamma _{t}},{\delta _{t}})$ satisfies all other (weaker) types of dependence discussed there. For instance, if F is DFR, then ${\gamma _{t}}$ and ${\delta _{t}}$ are associated, so that in particular $\mathit{Cov}({\gamma _{t}},{\delta _{t}})\ge 0$ for all t. Similarly, if F is IFR, the recurrence times are negatively associated, and thus $\mathit{Cov}({\gamma _{t}},{\delta _{t}})\le 0$ for all t. In fact, much less is required for the recurrence times to be (negatively) correlated. For, using Hoeffding’s identity (see Lehmann [11]), we get so that the covariance between ${\gamma _{t}}$ and ${\delta _{t}}$ is greater than (less than) zero provided that $({\gamma _{t}},{\delta _{t}})$ are positively (resp., negatively) quadrant dependent.

By Brown [4], if F is DFR, then and Recently, Losidis et al. [15] proved that if F is DFR, then In view of (6) and (7), Shaked and Zhu [21] proved the equivalence Motivated by the above discussion, in the next theorem we prove that the pair $({\gamma _{t}},{\delta _{t}})$ is increasing in the upper orthant order if and only if the renewal function $U(t)$ is concave.

Apart from the forward and backward recurrence times, another quantity of interest is their sum, ${\alpha _{t}}={\gamma _{t}}+{\delta _{t}}$, known as the total lifetime (or the spread), which is simply the length of the interarrival time including the current time t. Our next result concerns the stochastic monotonicity of the total lifetime. Recall that a variable X is smaller than Y in the increasing convex order (icx order), denoted by $X{\le _{\mathit{icx}}}Y$, if for any increasing, convex function ϕ (see, e.g., Shaked and Shanthikumar [20]).

Thus, for example, if the interarrival distribution F is DFR, then U is concave and from the corollary we have that ${\alpha _{t}}$ is increasing with respect to the icx order.

In the rest of this section, we focus on the bivariate tail of the joint distribution for the recurrence times, ${\gamma _{t}}$ and ${\delta _{t}}$. Extending recent work by Losidis and Politis [14] and Losidis et al. [15], who studied the covariance and the joint tail of these variables, we discuss monotonicity properties of this joint tail and their link with aging properties of the interarrival distribution F.

The next result generalizes Brown’s result in (6). The result has also been obtained, by different arguments, in Losidis et al. [15].

An interesting question is whether the renewal density $u(t)$ is monotone, decreasing or increasing, in $t\ge 0$. By (7), we see that if the distribution F is DFR, then $u(t)$ is a decreasing function in $t\ge 0$. However, if F is IFR, this does not imply that $u(t)$ is an increasing function (see, e.g., Shaked and Zhu [21]). Assuming that the renewal density $u(t)$ is an increasing function, we can see that $\mathbb{P}({\gamma _{t}}\gt y,{\delta _{t}}\gt x)$ is decreasing in t.

In the following result, we present a formula for the joint tail $\mathbb{P}({\gamma _{t}}\gt y,{\delta _{t}}\gt x)$. Using that formula we will be able to present conditions under which $\mathit{Cov}({\gamma _{t}},{\delta _{t}})\ge (\le )\hspace{0.1667em}0$. More specifically:

Proposition 4.4 could be used to determine conditions under which we have $\mathbb{P}({\gamma _{t}}\gt y,{\delta _{t}}\gt x)\ge (\le )\hspace{0.1667em}\mathbb{P}({\gamma _{t}}\gt y)\hspace{0.1667em}\mathbb{P}({\delta _{t}}\gt x)$ which, in view of the definitions in Section 2, means that the pair $({\gamma _{t}},{\delta _{t}})$ is PQD (NQD). More specifically, from Proposition 4.4 we obtain immediately the following two corollaries.

In the literature, the joint distribution of the backward and forward recurrence times and the right tails of those quantities have been studied extensively. However, relatively little attention has been paid to the conditional distribution of one of these quantities, given some information on the other. This gives us, in particular, additional insight into the dependence structure and the covariance between the two recurrence times. In this section we are focusing on the following two types of events.

1. At least x time units have passed since the last renewal, given that the next renewal will occur after y time units. In this case, we study the conditional probability $\mathbb{P}({\delta _{t}}\gt x\hspace{0.1667em}|\hspace{0.1667em}{\gamma _{t}}\gt y)$.

2. No renewal will occur during the next y time units, given that more than x time units have passed since the last revival. Here we look at the probability $\mathbb{P}({\gamma _{t}}\gt y\hspace{0.1667em}|\hspace{0.1667em}{\delta _{t}}\gt x)$.

Next, we present a formula for the two conditional probabilities above. Using these expressions we will be able to present conditions under which $\mathit{Cov}({\gamma _{t}},{\delta _{t}})\ge (\le )\hspace{0.1667em}0$. More specifically:

Next, we note that for any $x,y,t\ge 0$, see, e.g., Janssen and Manca [10, p. 82]. By using this, we see that and where in the last equality we use that $\mathbb{P}({\delta _{t+x}}\gt x)=\mathbb{P}({\gamma _{t}}\gt x)$; this follows by formula (12) or formula (13), for $y=0$ (see also Janssen and Manca [10, p. 79]). It is worth mentioning that the conditional probability in (14), is (in some sense) the residual distribution of ${\gamma _{t}}$. In the next theorem we prove that the probability $\mathbb{P}({\delta _{t+x}}\gt x\hspace{0.1667em}|\hspace{0.1667em}{\gamma _{t+x}}\gt y)$ is increasing in t, under the assumption that the interarrival times are DFR, generalizing Brown’s result in (6).

We now look at the other direction; the converse of Brown’s result in (7), namely that the concavity of U implies DFR interarrival times, is known as Brown’s conjecture and was disproved by Yu [23]. However, Chen [5] proved a weaker result: if U is concave, then the interarrival distribution F is NWU. He also showed that if U is convex, then F is NBU. To this end, we offer the following generalization of Chen’s result.

To illustrate the findings of the previous sections, we now give two examples; in the first one, the distribution F is DFR, in the second one it is IFR.