2. The distribution of the minimum observation until a stopping time

In a probability space $(\Omega,{\mathcal A},\mathbb{P})$ , let $X_{1},X_{2},\ldots$ be a sequence of independent and identically distributed (i.i.d.) random variables (RVs) with common cumulative distribution function (CDF) F. Moreover, consider an almost surely (a.s.) finite stopping time $T\in\{1,2,\ldots\}$ associated with this sequence. Remember that a discrete RV $T\in\{1,2,\ldots\}$ is a stopping time with respect to $X_{1},X_{2},\ldots$ when, for every $k\in\{1,2,\ldots\}$ , the event $[T=k]$ depends only on $X_{1},X_{2},\ldots,X_{k}$ (i.e. it is independent of $X_{k+1},X_{k+2},\ldots$ ). This implies that the indicator RV $\boldsymbol{1}_{[T\leq k]}$ is a function of $X_{1},X_{2},\ldots,X_{k}$ , i.e. $\boldsymbol{1}_{[T\leq k]}=g_{k}(X_{1},X_{2},\ldots,X_{k})$ for some measurable function $g_{k}$ . We are interested in the distribution of the minimum RV $M_{T}\;:\!=\;\min\{X_{1},X_{2},\ldots,X_{T}\}$ . Note that this distribution cannot be easily found by conditioning on the stopping time T because, in general, the $X_{i}$ are not independent of T.

In the same measurable space $(\Omega,{\mathcal A})$ , for $w\in\mathbb{R}$ such that $F(w)<1$ , we also consider a probability measure $\mathbb{P}_{w}$ such that $\mathbb{P}_{w}(A)\;:\!=\;\mathbb{E}\big(\boldsymbol{1}_{A}\varDelta_{n}^{w}\big)$ for every $A\in\mathcal{A}_{n}=\sigma(X_{1},\ldots,X_{n})$ , $n=1,2,\ldots$ , where

\begin{align*}\varDelta_{n}^{w}\;:\!=\;\prod_{i=1}^{n}\frac{\boldsymbol{1}_{[X_{i}>w]}}{1-F(w)},\quad n=1,2,\ldots\end{align*}

is a martingale with respect to the filtration $\mathcal{A}_{n}$ , $n=1,2,\ldots$ . Then, the RVs $X_{1},X_{2},\ldots$ are again i.i.d. under the w-truncated probability measure $\mathbb{P}_{w}$ , and each of them has the following w-truncated CDF $F_{w}$ :

(1) \begin{align} F_{w}(x) & \;:\!=\; \mathbb{P}_{w}(X_{n}\leq x) \nonumber \\ & = \mathbb{E}\big(\boldsymbol{1}_{[X_{n}\leq x]}\varDelta_{n}^{w}\big) = \mathbb{E}\bigg(\boldsymbol{1}_{[X_{n}\leq x]}\frac{\boldsymbol{1}_{[X_{n}>w]}}{1-F(w)}\bigg) = \frac{F(x)-F(w)}{1-F(w)}\boldsymbol{1}_{[x>w]}, \end{align}

since $X_1,X_2,\ldots,X_n$ are independent and $\mathbb{E}(\varDelta_{n}^{w})=1$ for all $n=1,2,\ldots$ and $w\in\mathbb{R}$ with $F(w)<1$ .

In what follows, we shall denote by $\mathbb{E}_{w}$ the expectation under $\mathbb{P}_{w}$ . So, under the above setup, we derive the following lemma.

Lemma 1. For any real measurable function h in $\mathbb{R}^{k}$ , $k\geq1$ , and $w\in\mathbb{R}$ such that $F(w)<1$ ,

\begin{align*} \mathbb{E}_{w}(h(X_{1},\ldots,X_{k})) = \frac{\mathbb{E}\big(\boldsymbol{1}_{[\!\min\{X_{1},X_{2},\ldots,X_{k}\}>w]}\,h(X_{1},\ldots,X_{k})\big)}{(1-F(w))^{k}}, \end{align*}

where $\mathbb{E}$ , $\mathbb{E}_{w}$ denote the expectations under the probability measures $\mathbb{P}$ , $\mathbb{P}_{w}$ , respectively.

Proof. Invoking the independence of $X_{1},\ldots,X_{k}$ in combination with (1), we have, for any measurable function h,

\begin{align*} \mathbb{E}_{w}(h(X_{1},\ldots,X_{k})) & = \int\cdots\int h(x_{1},\ldots,x_{k})\,{\mathrm{d}} F_{w}(x_{1}) \cdots {\mathrm{d}} F_{w}(x_{k}) \\ & = \int\cdots\int h(x_{1},\ldots,x_{k}) \frac{\boldsymbol{1}_{[x_{1}>w]}\cdots\boldsymbol{1}_{[x_{k}>w]}}{(1-F(w))^{k}}\,{\mathrm{d}} F(x_{1})\cdots{\mathrm{d}} F(x_{k}) \\ & = \frac{\mathbb{E}\big(\boldsymbol{1}_{[\!\min\{X_{1},X_{2},\ldots,X_{k}\}>w]}\,h(X_{1},\ldots,X_{k})\big)} {(1-F(w))^{k}}. \end{align*}

Now, the distribution of $M_{T}=\min\{X_{1},X_{2},\ldots,X_{T}\}$ may be obtained by exploiting the previous lemma, as shown in the result that follows.

Theorem 1. Let T be an a.s. finite stopping time associated with a sequence of i.i.d. RVs $X_{1},X_{2},\ldots$ with common CDF F. If $M_{T}=\min\{X_{1},X_{2},\ldots,X_{T}\}$ then

\begin{align*} \mathbb{P}(M_{T}>w) = \mathbb{E}_{w}\big((1-F(w))^{T}\big) \end{align*}

for $w\in\mathbb{R}$ such that $F(w)<1$ .

Proof. Since we have assumed that $P(T<\infty)=1$ , we have

\begin{align*} \mathbb{P}(M_{T}>w) = \sum_{k=1}^{\infty}\mathbb{P}(M_{T}>w,T=k) & = \sum_{k=1}^{\infty}\mathbb{E}\big(\boldsymbol{1}_{[T=k]}\boldsymbol{1}_{[M_{T}>w]}\big) \\ & = \sum_{k=1}^{\infty}\mathbb{E}\big(\boldsymbol{1}_{[T=k]}\boldsymbol{1}_{[\!\min\{X_{1},X_{2},\ldots,X_{k}\}>w]}\big) \\ & = \sum_{k=1}^{\infty}(1-F(w))^{k}\mathbb{E}_{w}\big(\boldsymbol{1}_{[T=k]}\big) \\ & = \sum_{k=1}^{\infty}(1-F(w))^{k}\mathbb{P}_{w}(T=k) = \mathbb{E}_{w}\big((1-F(w))^{T}\big), \end{align*}

where the fourth equality follows from a direct application of Lemma 1.

It is worth pointing out that Lemma 1 and Theorem 1 do not rely on the existence of a density for the RVs $X_{1},X_{2},\ldots$ Furthermore, it should be noted that Theorem 1 is valid generally for every stopping time under the above assumptions. However, in the rest of this paper we shall be mainly concerned with a specific type of stopping time associated with the number of arrivals in a renewal process in a given time interval. Applications considering other types of stopping times will be the purpose of subsequent work.

3. The distribution of the minimal spacing in a renewal process

Under the setup of Section 2, we assume that the i.i.d. RVs $X_{1},X_{2},\ldots$ are the interarrival times of a renewal process $\{Y_{t},t\geq0\}$ , i.e. $Y_{t}=\sup\{i\geq1\colon X_{1}+\cdots+X_{i}\leq t\}$ . Moreover, we set $T\;:\!=\;Y_{t}+1$ . Then T is the number of summands until the sum of the $X_{i}$ exceeds t for the first time. Also, the event $[Y_{t}+1=k]$ depends only on $X_{1},X_{2},\ldots,X_{k}$ , and therefore T is a stopping time associated with the sequence $X_{1},X_{2},\ldots$ . Thus, the RV $M_{T}\;:\!=\;\min\{X_{1},X_{2},\ldots,X_{T}\}=\min\{X_{1},X_{2},\ldots,X_{Y_{t}+1}\}$ will now denote the minimal spacing that starts in the interval [0, t]. Here, notice that if $S_{k}\;:\!=\;X_{1}+\cdots+X_{k}$ , $k=1,2,\ldots$ , is the kth arrival epoch of the renewal process $\{Y_{t},t\geq0\}$ , then the last spacing may extend beyond t because $S_{Y_{t}}\leq t$ and $S_{Y_{t}+1}\geq t$ .

Throughout the rest of this paper, since T is a function of t, the minimal spacing will be denoted by $D_{t}$ instead of $M_{T}$ to emphasize its dependence on time t. Notice also that the truncated measure $\mathbb{P}_{w}$ will be meaningfully defined only for $w\geq0$ such that $F(w)<1$ since the RVs $X_{1},X_{2},\ldots$ will be non-negative (also with finite expectation) as being the interarrival times of a renewal process.

The next result provides a general expression for the distribution of the minimal spacing $D_{t}$ in a renewal process.

Theorem 2. Let $D_{t}=\min\{X_{1},X_{2},\ldots,X_{T}\}$ be the minimal spacing that starts in the time interval [0, t] in a counting renewal process $\{Y_{t},t\geq0\}$ , where $T=Y_{t}+1$ and $X_{1},X_{2},\ldots$ are the interarrival spacings between successive appearances of events with common CDF F. Then

\begin{equation*} \mathbb{P}(D_{t} > w) = \begin{cases} {\displaystyle F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\big(1-F_{w}^{*k}(t)\big),} & 0 < w < t, \\ 1-F(w), & w\geq t, \end{cases} \end{equation*}

where $S_{k}=\sum_{i=1}^{k}X_{i}$ and $F_{w}^{*k}(t) \;:\!=\; \mathbb{P}(S_{k}\leq t \mid X_{i}>w$ for all $i=1,\ldots,k)$ denotes the CDF of the kth-order convolution of the w-truncated F.

Proof. Applying Theorem 1, we get

\begin{align*} \mathbb{P}(D_{t} > w) & = \mathbb{E}_{w}\big((1-F(w))^{T}\big) \\ & = \sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}_{w}(T=k) \\ & = \sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}_{w}(T\leq k) - \sum_{k=2}^{\infty}(1-F(w))^{k}\,\mathbb{P}_{w}(T\leq k-1) \\ & = \sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}_{w}(T\leq k) - \sum_{k=1}^{\infty}(1-F(w))^{k+1}\mathbb{P}_{w}(T\leq k) \\ & = F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}_{w}(T\leq k) \end{align*}

(obviously, $\mathbb{P}_{w}(T\leq0)=0$ ), and taking into account that $[Y_{t}\leq k-1]=[S_{k}>t]$ , we finally obtain that

\begin{equation*} \mathbb{P}_{w}(T\leq k) = \mathbb{P}_{w}(Y_{t}\leq k-1) = \mathbb{P}_{w}(S_{k}>t) = \mathbb{P}(S_{k}>t \mid X_{i}>w\ \text{for all}\ i=1,\ldots,k). \end{equation*}

Notice that $\mathbb{P}(S_{k}>t \mid X_{i}>w\ \text{for all}\ i=1,\ldots,k)=1$ when $t\leq w$ , which in this case yields

\begin{align*} \mathbb{P}(D_{t} > w) = F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}=1-F(w), \end{align*}

and the proof is completed.

Clearly, in order to find a more explicit formula for $\mathbb{P}(D_{t} > w)$ by Theorem 2, we should have at hand the exact distribution of the arrival epochs $S_{k}$ . As we will see in Proposition 1, if we consider a Poisson process then an exact formula for $\mathbb{P}(D_{t} > w)$ can be derived by this result.

Next, aside from the above-mentioned (dependent on the $X_{i}$ ) renewal process $\{Y_{t},t\geq0\}$ , we also consider another renewal process $\{K_{t},t\geq0\}$ with $K_{t}=_{{\mathrm{st}}}Y_{t}$ for all $t\geq0$ , where ‘ $=_{{\mathrm{st}}}$ ’ denotes equality in distribution. In addition, the process $\{K_{t},t\geq0\}$ is assumed to be independent of the $X_{i}$ . If $G_{t}\;:\!=\;\min\{X_{1},X_{2},\ldots,X_{K_{t}+1}\}$ , then we shall prove that the minimal spacing $D_{t}$ is larger stochastically than $G_{t}$ with respect to the usual stochastic ordering (cf. [Reference Shaked and Shanthikumar33, Reference Szekli36]).

We then obtain the following result, which implies that, when the dependence between the spacings and the stopping time is ignored, we get a stochastically smaller minimum spacing.

Proof. Theorem 2 yields, for $w>0$ ,

(2) \begin{align} \mathbb{P}(D_{t} > w) & = F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\, \mathbb{P}(S_{k} > t \mid X_{i} > w\ \text{for all}\ i=1,\ldots,k) \nonumber \\ & = F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\, \mathbb{P}(S_{k} > t \mid \min\{X_{1},\ldots,X_{k}\} > w) \end{align}

(it can be easily seen that the previous formula includes both cases $0 < w < t$ and $w\geq t$ , which are stated separately in the aforementioned theorem). Exploiting the fact that $S_{k}=\sum_{i=1}^{k}X_{i}$ and $(X_{1},\ldots,X_{k})$ are positively associated (cf. [Reference Barlow and Proschan9]), or equivalently that $S_{k}$ and $\min\{X_{1},\ldots,X_{k}\}$ are positively quadrant dependent (cf. [Reference Lehmann26]), we obtain that

\begin{align*} \mathbb{P}(S_{k}>t,\min\{X_{1},\ldots,X_{k}\}>w) \geq \mathbb{P}(S_{k}> t)\mathbb{P}(\!\min\{X_{1},\ldots,X_{k}\}>w), \end{align*}

and thus

\begin{equation*} \mathbb{P}(S_{k}>t \mid \min\{X_{1},\ldots,X_{k}\}>w) \geq \mathbb{P}(S_{k}>t) = \mathbb{P}(Y_{t}\leq k-1) = \mathbb{P}(K_{t}\leq k-1), \end{equation*}

since $K_{t}=_{{\mathrm{st}}}Y_{t}$ . Hence, combining this with (2), we conclude that

(3) \begin{equation} \mathbb{P}(D_{t} > w) \geq F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}(K_{t}\leq k-1), \quad w > 0. \end{equation}

On the other hand, a direct application of the total probability theorem yields, for $w>0$ ,

(4) \begin{align} \mathbb{P}(G_{t} > w) & = \sum_{k=0}^{\infty}\mathbb{P}(\!\min\{X_{1},X_{2},\ldots,X_{K_{t}+1}\} > w \mid K_{t}=k) \mathbb{P}(K_{t}=k) \nonumber \\ & = \sum_{k=0}^{\infty}\mathbb{P}(\!\min\{X_{1},X_{2},\ldots,X_{k+1}\} > w)\mathbb{P}(K_{t}=k) \nonumber \\ & = \sum_{k=0}^{\infty}(1-F(w))^{k+1}(\mathbb{P}(K_{t}\leq k)-\mathbb{P}(K_{t}\leq k-1)) \nonumber \\ & = F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}(K_{t}\leq k-1), \end{align}

and therefore we finally get that $\mathbb{P}(D_{t} > w) \geq \mathbb{P}(G_{t} > w)$ for all $w>0$ by virtue of (3) and (4).

In the proof of Theorem 3 it was shown that

\begin{align*} \mathbb{P}(D_{t} > w) \geq \mathbb{P}(G_{t} > w) = F(w)\sum_{k=1}^{\infty}(1-F(w))^{k}\,\mathbb{P}(S_{k} > t).\end{align*}

An interesting remark here is that the lower bound $\mathbb{P}(G_{t} > w)$ may equivalently be expressed as $\mathbb{P}(G_{t} > w)=\mathbb{P}(S_{Q}>t)$ , where Q follows the geometric distribution with parameter F(w) and mass function $\mathbb{P}(Q=k)=F(w)(1-F(w))^{k}$ for $k=0,1,\ldots$ , and, in addition, Q is independent of the $X_{i}$ , so that $S_{Q}=X_{1}+\cdots+X_{Q}$ is a geometric convolution (with $S_{0}=0$ ). Concerning that, [Reference Brown14, Theorem 2.1] provides a lower bound for the probability $\mathbb{P}(S_{Q}>t)=\mathbb{P}(G_{t} > w)$ which, under our setup, yields

\begin{equation*} \mathbb{P}(D_{t} > w) \geq (1-F(w))\exp\bigg\{{-}\frac{F(w)}{1-F(w)}\bigg(\frac{t}{\mathbb{E}[X_{1}]}+\frac{\mathbb{E}[X_{1}^{2}]}{\mathbb{E}[X_{1}]^{2}}-1\bigg)\bigg\}.\end{equation*}

Moreover, in the special case where the distribution of $X_{i}$ possesses the ‘new better than used in expectation’ (NBUE) property [Reference Barlow and Proschan9, Reference Shaked and Shanthikumar33], [Reference Brown14, (3.1.10)] yields an improved lower bound for the probability $\mathbb{P}(S_{Q}>t)=\mathbb{P}(G_{t} > w)$ which, in our case, gives

(5) \begin{equation} \mathbb{P}(D_{t} > w) \geq (1-F(w))\exp\bigg\{{-}\frac{t\,F(w)}{(1-F(w))\mathbb{E}[X_{1}]}\bigg\} \;=\!:\; \mathit{LB}_{_{\mathrm{Br}}}(w).\end{equation}

In what follows, we mainly focus on applications where the $X_{i}$ have an absolutely continuous distribution (e.g. exponential, gamma, uniform). However, it should be noted that Theorems 2 and 3 as well as Proposition 2 are valid regardless of the existence of a density for the distribution of the $X_{i}$ .

Next, we consider a Poisson counting process $\{N_{t},t\geq0\}$ with constant intensity $\lambda\in(0,+\infty)$ , which is a special case of a renewal process. As is already known, in a Poisson process the interarrival times $X_{1},X_{2},\ldots$ follow the exponential distribution $\mathcal{E}(\lambda)$ with parameter $\lambda\in(0,+\infty)$ , and the kth arrival epochs $S_{k}=\sum_{i=1}^{k}X_{i}$ have the Erlang (gamma) distribution with scale parameter $\lambda$ and shape parameter k, to be denoted by $\mathcal{G}(k,\lambda)$ .

As a consequence of Theorem 3, we immediately derive the next result for a Poisson counting process.

We are also interested in the exact distribution of the minimal spacing $D_{t}$ when dealing with a Poisson process. In the next proposition, an explicit formula is provided for the tail probability $\mathbb{P}(D_{t} > w)$ . The key point for its proof lies in Theorem 2 and the loss of memory of the exponential distribution.

In what follows, for any RV X and an event A, by $[X\mid A]$ we shall denote an RV whose distribution is the conditional distribution of X given A. We shall also use the standard notation $\Gamma(a,z)\;:\!=\;\int_{z}^{\infty}x^{a-1}{\mathrm{e}}^{-x}\,{\mathrm{d}} x$ , with $a>0$ and $z\geq0$ , for the incomplete gamma function, where $\Gamma(k)=\Gamma(k,0)=(k-1)!$ for $k=0,1,2,\ldots$

Proposition 1. Let $\{N_{t},t\geq0\}$ be a Poisson process with intensity $\lambda\in(0,+\infty)$ , and $X_{1},X_{2},\ldots$ the interarrival spacings between successive appearances of events. If $T=N_{t}+1$ , then the distribution of the minimal spacing $D_{t}=\min\{X_{1},X_{2},\ldots,X_{T}\}$ that starts in the time interval [0, t] is given by

\begin{equation*} \mathbb{P}(D_{t} > w) = \begin{cases} {\displaystyle (1-{\mathrm{e}}^{-\lambda w})\sum_{k=1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk} \frac{\Gamma(k,\lambda(t-kw))}{(k-1)!} + {\mathrm{e}}^{-\lambda w(\lfloor{t}/{w}\rfloor+1)},} & 0 < w < t, \\ {\mathrm{e}}^{-\lambda w}, & w\geq t, \end{cases} \end{equation*}

where $\lfloor{t}/{w}\rfloor$ is the integer part of ${t}/{w}$ .

Proof. Applying Theorem 2 combined with the fact that the $X_{i}$ are exponential distributed with parameter $\lambda$ (and thus $F(w)=1-{\mathrm{e}}^{-\lambda w}$ , $w\geq0$ ), we immediately conclude that

(6) \begin{equation} \mathbb{P}(D_{t} > w) = (1-{\mathrm{e}}^{-\lambda w})\sum_{k=1}^{\infty}{\mathrm{e}}^{-\lambda wk} \mathbb{P}(S_{k} > t \mid X_{i} > w\ \text{for all}\ i=1,\ldots,k), \quad 0 < w < t, \end{equation}

and $\mathbb{P}(D_{t} > w)=1-F(w)={\mathrm{e}}^{-\lambda w}$ for $w\geq t$ . Exploiting the loss of memory of the exponential distribution, i.e. $[X_{i}\mid X_{i}>w] =_{\mathrm{st}} X_{i}+w$ , we obtain that

(7) \begin{align} \mathbb{P}(S_{k}\geq t \mid X_{i}>w\ \text{for all}\ i=1,2,\ldots,k) & = \mathbb{P}(S_{k}\geq t-kw) \nonumber \\ & = \begin{cases} \dfrac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)}, & 1\leq k\leq{t}/{w}, \\ 1, & k>{t}/{w}, \end{cases} \end{align}

since, in this case, $S_{k}$ follows the gamma distribution $\mathcal{G}(k,\lambda)$ . Hence, for $t>w$ , (6) in virtue of (7) yields

\begin{align*} \mathbb{P}(D_{t} > w) & = (1-{\mathrm{e}}^{-\lambda w})\Bigg(\sum_{k=1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk} \frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)} + \sum_{k=\lfloor{t}/{w}\rfloor+1}^{\infty}{\mathrm{e}}^{-\lambda wk} \times 1\Bigg) \\ & = (1-{\mathrm{e}}^{-\lambda w})\Bigg(\sum_{k=1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk} \frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)} + \frac{{\mathrm{e}}^{-\lambda w(\lfloor{t}/{w}\rfloor+1)}}{1-{\mathrm{e}}^{-\lambda w}}\Bigg) \\ & = (1-{\mathrm{e}}^{-\lambda w})\sum_{k=1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk} \frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)} + {\mathrm{e}}^{-\lambda w(\lfloor{t}/{w}\rfloor+1)}, \end{align*}

and the proof is complete.

It is well known that

(8) \begin{equation} \frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)} = {\mathrm{e}}^{-\lambda(t-kw)}\sum_{i=0}^{k-1}\frac{\lambda^{i}(t-kw)^{i}}{i!},\end{equation}

and therefore the tail probability of $D_{t}$ in Proposition 1 can be equivalently written as

\begin{align*} \mathbb{P}(D_{t} > w) = (1-{\mathrm{e}}^{-\lambda w}){\mathrm{e}}^{-\lambda t}\sum_{k=1}^{\lfloor{t}/{w}\rfloor}\sum_{i=0}^{k-1} \frac{\lambda^{i}(t-kw)^{i}}{i!} + {\mathrm{e}}^{-\lambda w(\lfloor{t}/{w}\rfloor+1)}\end{align*}

for $0 < w < t$ .

Recalling (7) in the proof of Proposition 1, the equality

(9) \begin{equation} \mathbb{P}(S_{k}\geq t \mid X_{i}>w\ \text{for all}\ i=1,2,\ldots,k) = \mathbb{P}(S_{k}\geq t-kw)\end{equation}

enabled us to express the probability $\mathbb{P}(D_{t}> w)$ directly through the distribution of $S_{k}$ . This relation is a consequence of the memoryless property of the exponential distribution, which implies that $[X_{i}\mid X_{i}>w]=_{\mathrm{st}}X_{i}+w$ .

Apart from the exponential and geometric distributions, no other cases exist where (9) is valid. However, there are classes of distributions where we may get inequality instead of equality in (9). In this respect, we note the definition of the well-known IFR and DFR classes of distributions (cf. [Reference Barlow and Proschan9, Reference Shaked and Shanthikumar33]).

Exploiting [Reference Shaked and Shanthikumar33, Theorem 1.A.30(a)], we get the following stochastic relations for non-negative IFR and DFR RVs by means of the usual stochastic order. Specifically, for a non-negative IFR (DFR) RV X,

(10) \begin{equation} [X \mid X>w] \preceq_{\mathrm{st}} (\succeq_{\mathrm{st}})\,X+w.\end{equation}

We note that the exponential distribution is both IFR and DFR.

Considering all the above, we derive the following general result for the survival function of the minimal spacing $D_{t}$ in a renewal process with IFR or DFR interarrival times.

Proposition 2. If the interarrival spacings $X_{1},X_{2},\ldots$ between successive appearances of events in a renewal process are IFR (DFR), then

\begin{equation*} \mathbb{P}(D_{t} > w) \leq (\geq)\,F(w)\sum_{k=1}^{\lfloor{t}/{w}\rfloor}(1-F(w))^{k} \mathbb{P}(S_{k} > t-kw) + (1-F(w))^{\lfloor{t}/{w}\rfloor+1}, \quad 0 < w < t, \end{equation*}

where $S_{k}=\sum_{i=1}^{k}X_{i}$ . Equality holds when the $X_{i}$ are exponentially distributed.

Proof. Under the assumption that the $X_{i}$ are IFR (DFR), property (10) and the fact that the usual stochastic order is closed under convolutions together yield

\begin{align*} \mathbb{P}(S_{k}\geq t \mid X_{i}>w\ \text{for all}\ i=1,2,\ldots,k) \leq (\geq)\,\mathbb{P}(S_{k}\geq t-kw) \end{align*}

and the result follows from Theorem 2, similarly to the proof of Proposition 1.

It can be easily verified that the lower bound of Proposition 2 is always better (larger) than that of Theorem 3 (since $\mathbb{P}(S_{k}>t)\leq\mathbb{P}(S_{k}>t-kw)$ ), a fact that is intuitively expected since the former bound refers to a smaller class of distributions that possess the DFR property, whereas the latter one is more general.

Example 1. As an example, we assume that the spacings $X_{i}$ follow the Gamma distribution $\mathcal{G}(a,\lambda)$ with density $f(x)=({\lambda^{a}}/{\Gamma(a)})x^{a-1}{\mathrm{e}}^{^{-\lambda x}}\boldsymbol{1}[x>0]$ , which is IFR when $a\geq1$ and DFR when $0 < a\leq1$ [Reference Marshall and Olkin27]. Since, in this case, $S_{k}=\sum_{i=1}^{k}X_{i}$ has the gamma distribution $\mathcal{G}(ka,\lambda)$ , Proposition 2 directly yields, for $a>1$ ( $0 < a < 1$ ) and $0 < w < t$ ,

(11) \begin{align} \mathbb{P}(D_{t} > w) & \leq (\geq)\,\bigg(1-\frac{\Gamma(a,\lambda w)}{\Gamma(a)}\bigg) \sum_{k=1}^{\lfloor{t}/{w}\rfloor}\bigg(\frac{\Gamma(a,\lambda w)}{\Gamma(a)}\bigg)^{k} \frac{\Gamma(ka,\lambda(t-kw))}{\Gamma(ka)} \nonumber \\ & \qquad\quad + \bigg(\frac{\Gamma(a,\lambda w)}{\Gamma(a)}\bigg)^{(\lfloor{t}/{w}\rfloor+1)}, \end{align}

whereas for $w\geq t$ , $\mathbb{P}(D_{t} > w) = \Gamma(a,\lambda w)/\Gamma(a)$ for all $a>0$ .

Next, we return to the case of a Poisson process. Aside from the explicit formula of Proposition 1 for the exact distribution of $D_{t}$ , it would be quite useful to have at hand easily computable closed-form upper and lower bounds for the tail probability $\mathbb{P}(D_{t} > w)$ that can also lead to asymptotic results. We offer the following proposition.

Proposition 3. Consider a Poisson process $\{N_{t},t\geq0\}$ with intensity $\lambda\in(0,+\infty)$ and interarrival spacings $X_{1},X_{2},\ldots$ between successive appearances of events. If $T=N_{t}+1$ and $D_{t}=\min\{X_{1},X_{2},\ldots,X_{T}\}$ , then for $t>w>0$ and $\lambda t>1$ , $L(\lambda,t,w)\mathbb{\leq P}(D_{t} > w)\leq U(\lambda,t,w)$ , where

\begin{align*} L(\lambda,t,w) & \;:\!=\; \exp\{-\lambda w-\lambda t(1-{\mathrm{e}}^{-\lambda w})\}, \\ U(\lambda,t,w) & \;:\!=\; \exp\bigg\{{-}\lambda w - \lambda\bigg(t-w\bigg\lfloor\frac{1}{w}\sqrt{\frac{t}{\lambda}}\bigg\rfloor\bigg)(1-{\mathrm{e}}^{-\lambda w})\bigg\} + \exp\bigg\{{-}\lambda w\bigg(\bigg\lfloor\frac{1}{w}\sqrt{\frac{t}{\lambda}}\bigg\rfloor+1\bigg)\bigg\}. \end{align*}

Proof. Applying Proposition 1 for $w<t$ , we get

(12) \begin{equation} \mathbb{P}(D_{t} > w) = c(\lambda,t,w) + {\mathrm{e}}^{-\lambda w(\lfloor{t}/{w}\rfloor+1)}, \end{equation}

where, for convenience, we set

\begin{align*} c(\lambda,t,w) \;:\!=\; (1-{\mathrm{e}}^{-\lambda w})\sum_{k=1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk} \frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)}. \end{align*}

The summand $c(\lambda,t,w)$ will lead us to the desired upper bound, and hence we will treat it separately. So, recalling that $\Gamma(k,\lambda(t-kw))/\Gamma(k)=\mathbb{P}(S_{k}\geq t-kw)\leq1$ , and taking into consideration that $\Gamma(k,\lambda(t-kw))\leq\Gamma(k,\lambda(t-bw))$ when $k\leq b\;:\!=\;\lfloor({1}/{w})\sqrt{{t}/{\lambda}}\rfloor$ , we obtain

(13) \begin{align} c(\lambda,t,w) & = (1-{\mathrm{e}}^{-\lambda w})\Bigg\{\sum_{k=1}^{b}{\mathrm{e}}^{-\lambda wk}\frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)} + \sum_{k=b+1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk}\frac{\Gamma(k,\lambda(t-kw))}{\Gamma(k)}\Bigg\} \nonumber \\ & \leq (1-{\mathrm{e}}^{-\lambda w})\Bigg\{\sum_{k=1}^{\infty}{\mathrm{e}}^{-\lambda wk}\frac{\Gamma(k,\lambda(t-bw))}{\Gamma(k)} + \sum_{k=b+1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk}\Bigg\} . \end{align}

As will become apparent, the splitting of the summation at b was chosen so that we get a simple-in-form upper bound that is asymptotically sharp (cf. the proof of Theorem 4). The first summation in (13) is now equal to

\begin{align*} \sum_{k=1}^{\infty}\frac{{\mathrm{e}}^{-\lambda wk}}{\Gamma(k)}\int_{\lambda(t-bw)}^{\infty}s^{k-1}{\mathrm{e}}^{-s}\,{\mathrm{d}} s & = \int_{\lambda(t-bw)}^{\infty}{\mathrm{e}}^{-s}\Bigg(\sum_{k=1}^{\infty}\frac{{\mathrm{e}}^{-\lambda wk}}{\Gamma(k)}s^{k-1}\Bigg)\,{\mathrm{d}} s \\ & = \frac{{\mathrm{e}}^{-\lambda w}\exp\{-\lambda(t-bw)(1-{\mathrm{e}}^{-\lambda w})\}}{1-{\mathrm{e}}^{-\lambda w}}, \end{align*}

where the interchange of summation and integration is allowed by Tonelli’s theorem. Therefore,

\begin{align*} c(\lambda,t,w) & \leq (1-{\mathrm{e}}^{-\lambda w})\Bigg\{\frac{{\mathrm{e}}^{-\lambda w}\exp\{-\lambda(t-bw)(1-{\mathrm{e}}^{-\lambda w})\}}{1-{\mathrm{e}}^{-\lambda w}} + \sum_{k=b+1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-\lambda wk}\Bigg\} \nonumber \\ & = {\mathrm{e}}^{-\lambda w}\exp\{-\lambda(t-bw)(1-{\mathrm{e}}^{-\lambda w})\} + {\mathrm{e}}^{-\lambda w(b+1)} - {\mathrm{e}}^{-\lambda w(\lfloor{t}/{w}\rfloor+1)}. \end{align*}

Combining this with (12), we finally obtain the upper bound

\begin{align*} \mathbb{P}(D_{t} > w) \leq {\mathrm{e}}^{-\lambda w}\exp\{-\lambda(t-bw)(1-{\mathrm{e}}^{-\lambda w})\} + {\mathrm{e}}^{-\lambda w(b+1)} \;=\!:\; U(\lambda,t,w). \end{align*}

The lower bound $L(\lambda,t,w)$ will be derived by Corollary 1. In particular, by a straightforward application of this corollary we deduce that $\mathbb{P}(G_{t} > w) \leq \mathbb{P}(D_{t} > w)$ , $w\in(0,\infty)$ . Invoking the second equality of (4) in the proof of Theorem 3, and recalling that $K_{t}$ has the Poisson distribution with parameter $\lambda t$ since $K_{t}=_{\mathrm{st}}N_{t}$ , we get

\begin{align*} \mathbb{P}(G_{t} > w) & = \sum_{k=0}^{\infty}\mathbb{P}(\!\min\{X_{1},X_{2},\ldots,X_{k+1}\} > w)P(K_{t}=k) \\ & = \sum_{k=0}^{\infty}{\mathrm{e}}^{-\lambda w(k+1)}{\mathrm{e}}^{-\lambda t}\frac{(\lambda t)^{k}}{k!} = \exp\{-\lambda w-\lambda t(1-{\mathrm{e}}^{-\lambda w})\}, \end{align*}

which is exactly the lower bound $L(\lambda,t,w)$ .

Although an exact formula for the distribution of $D_{t}$ in a Poisson process has been obtained in Proposition 1, it is also of theoretical as well as practical interest to investigate its asymptotic behaviour. In particular, it is shown below that, under general conditions, the minimum spacing $D_{t}$ satisfies an asymptotic result.

Proof. For $w={x}/(\lambda^{2}t)>0$ , Proposition 3 yields

(14) \begin{equation} L\bigg(\lambda,t,\frac{x}{\lambda^{2}t}\bigg) \leq \mathbb{P}(\lambda^{2}tD_{t}>x) \leq U\bigg(\lambda,t,\frac{x}{\lambda^{2}t}\bigg). \end{equation}

It can be easily verified that, for $x>0$ ,

(15) \begin{equation} \lim_{\lambda t\to\infty}L\bigg(\lambda,t,\frac{x}{\lambda^{2}t}\bigg) = \lim_{\lambda t\to\infty}\{{\mathrm{e}}^{-{x}/({\lambda t})}\exp\{-\lambda t\,(1-e^{-{x}/({\lambda t})})\}\} = {\mathrm{e}}^{-x}. \end{equation}

Moreover, the following limit relations also hold:

\begin{align*} \lim_{\lambda t\rightarrow\infty}\exp\bigg\{{-}\frac{x}{\lambda t} \bigg(\bigg\lfloor\frac{(\lambda t)^{3/2}}{x}\bigg\rfloor + 1\bigg)\bigg\} & = 0, \\ \lim_{\lambda t\rightarrow\infty}\bigg({\mathrm{e}}^{-{x}/({\lambda t})} \exp\bigg\{{-}\bigg(\lambda t-\bigg\lfloor\frac{(\lambda t)^{3/2}}{x}\bigg\rfloor \frac{x}{\lambda t}\bigg)(1-e^{-{x}/({\lambda t})})\bigg\}\bigg) & = {\mathrm{e}}^{-x}. \end{align*}

Consequently,

(16) \begin{equation} \lim_{\lambda t\to\infty}U\bigg(\lambda,t,\frac{x}{\lambda^{2}t}\bigg) = {\mathrm{e}}^{-x}. \end{equation}

Thus, combining (14) with (15) and (16), we finally obtain the desired approximation result.

The previous asymptotic outcome implies that, for large $\lambda t$ , it is approximately $\mathbb{P}(D_{t} > w )\simeq {\mathrm{e}}^{-w\lambda^{2}t}$ .

Note that in the aforementioned results we were mainly concerned with the minimum spacing in a Poisson process, due to the theoretical importance, applicability, and nice properties of this process. However, it is worth stressing that Theorem 2 is valid for every renewal process. Hence, in order to illustrate its wider applicability, we now present two additional results concerning the minimum spacing for specific non-exponential distributions of the interarrival times.

Initially, we shall extract an explicit formula for the survival function of $D_{t}$ when the interarrival times of the renewal process follow the Erlang distribution $\mathcal{G}(2,\lambda)$ .

Proposition 4. If the CDF of the interarrival times $X_{1},X_{2},\ldots$ of a renewal process is given by $F(x) = 1 - {\mathrm{e}}^{-\lambda x}(1+\lambda x)$ , $x\geq0$ , then, for $0 < w < t$ ,

\begin{align*} \mathbb{P}(D_{t} > w) & = (1-{\mathrm{e}}^{-\lambda w}(1+\lambda w)){\mathrm{e}}^{-\lambda t}\sum_{k=1}^{\lfloor{t}/{w}\rfloor}\sum_{\nu=0}^{k} \binom{k}{\nu}(\lambda w)^{\nu}\sum_{i=0}^{2k-\nu-1}\frac{\lambda^{i}(t-kw)^{i}}{i!} \\ & \quad + ({\mathrm{e}}^{-\lambda w}(1+\lambda w))^{\lfloor{t}/{w}\rfloor+1} \end{align*}

and $\mathbb{P}(D_{t} > w) = {\mathrm{e}}^{-\lambda w}(1+\lambda w)$ for $w\geq t$ .

Proof. In this case, the conditional distribution of $[X_{i}-w \mid X_{i}>w]$ is a mixture of an exponential $\mathcal{E}(\lambda)$ and an Erlang $\mathcal{G}(2,\lambda)$ distribution. Indeed, for $t>0$ and $w>0$ ,

\begin{align*} \mathbb{P}(X_{i}-w>t \mid X_{i}>w) = \frac{\mathbb{P}(X_{i}>t+w)}{\mathbb{P}(X_{i}>w)} & = \frac{{\mathrm{e}}^{-\lambda(t+w)}(1+\lambda(t+w))}{{\mathrm{e}}^{-\lambda w}(1+\lambda w)} \\ & = \frac{{\mathrm{e}}^{-\lambda t}(1+\lambda(t+w))}{(1+\lambda w)} \\ & = p{\mathrm{e}}^{-\lambda t} + (1-p){\mathrm{e}}^{-\lambda t}(1+\lambda t), \end{align*}

where $p={\lambda w}/({1+\lambda w})\in(0,1)$ . Therefore, $[X_{i}-w \mid X_{i}>w] =_{\mathrm{st}} I_{i}Z_{i}+(1-I_{i})R_{i}$ , where the RVs $I_{i}$ , $Z_{i}$ , and $R_{i}$ are all considered independent following a Bernoulli(p), an exponential $\mathcal{E}(\lambda)$ , and an Erlang $\mathcal{G}(2,\lambda)$ distribution, respectively. Then, it follows that

\begin{align*} & \mathbb{P}(S_{k}>t \mid X_{i}>w, i=1,\ldots,k) \\ & \qquad = \mathbb{P}\Bigg(\sum_{i=1}^{k}(I_{i}Z_{i}+(1-I_{i})R_{i})>t-kw\Bigg) \\ & \qquad = \sum_{\nu=0}^{k}\mathbb{P}\Bigg(\sum_{i=1}^{k}(I_{i}Z_{i}+(1-I_{i})R_{i}) > t-kw \mid \sum_{i=1}^{k}I_{i}=\nu\Bigg)\mathbb{P}\Bigg(\sum_{i=1}^{k}I_{i}=\nu\Bigg). \end{align*}

We observe that the conditional distribution of the RV $\sum_{i=1}^{k}(I_{i}Z_{i}+(1-I_{i})R_{i})$ given that $\sum_{i=1}^{k}I_{i}=\nu$ coincides with the distribution of $\sum_{i=1}^{\nu}Z_{i}+\sum_{i=1}^{k-\nu}R_{i}$ , which, as a convolution of exponential and Erlang distributions, also follows the Erlang distribution with scale parameter $\lambda$ and shape parameter $\nu+2(k-\nu)=2k-\nu$ . Hence,

\begin{align*} \mathbb{P}(S_{k}>t \mid X_{i}>w, i=1,\ldots,k) & =\sum_{\nu=0}^{k}\mathbb{P}\Bigg(\sum_{i=1}^{\nu}Z_{i}+\sum_{i=1}^{k-\nu}R_{i}>t-kw\Bigg)\binom{k}{\nu} p^{\nu}(1-p)^{k-\nu} \\ & = \sum_{\nu=0}^{k}\binom{k}{\nu}(\lambda w)^{\nu} \frac{\Gamma(2k-\nu,\lambda(t-kw))}{\Gamma(2k-\nu)(1+\lambda w)^{k}} \end{align*}

and a direct application of Theorem 2 yields, for $0 < w < t$ ,

\begin{align*} \mathbb{P}(D_{t} > w) & = F(w)\sum_{k=1}^{\lfloor{t}/{w}\rfloor}(1-F(w))^{k}\mathbb{P}(S_{k}>t \mid X_{i}>w, i=1,\ldots,k) + (1-F(w))^{\lfloor{t}/{w}\rfloor+1} \\ & = (1-{\mathrm{e}}^{-\lambda w}(1+\lambda w))\sum_{k=1}^{\lfloor{t}/{w}\rfloor}{\mathrm{e}}^{-k\lambda w}\sum_{\nu=0}^{k} \binom{k}{\nu}(\lambda w)^{\nu}\frac{\Gamma(2k-\nu,\lambda(t-kw))}{\Gamma(2k-\nu)} \\ & \quad + ({\mathrm{e}}^{-\lambda w}(1+\lambda w))^{\lfloor{t}/{w}\rfloor+1}, \end{align*}

which, also taking (8) into account, finally completes the proof.

Before closing this section, we present an additional application of Theorem 2, this time assuming that the spacings $X_{1},X_{2},\ldots$ follow the uniform distribution in (0,1), to be denoted by $\mathcal{U}(0,1)$ . In addition, we shall exploit the fact that the convolution of uniform distributions converges quickly to a normal distribution by the central limit theorem (CLT), allowing us to employ a normal approximation for the summands when t is relatively large (e.g. greater than 3 will suffice; see Figure 3). In what follows, $\mathrm{sgn}(x)$ denotes the sign of x.

4. Numerical examples

Next, we present several numerical comparisons among the exact distribution of $D_{t}$ , the upper/lower bounds, and the approximate formulae presented in Section 3. We begin by assuming that the events appear according to a homogeneous Poisson process with constant intensity $\lambda$ , i.e. the spacings are exponentially distributed. The exact tail distribution of the minimal spacing $D_{t}$ that starts in a time interval [0, t] is given by Proposition 1 and is depicted for explicit values of the parameters in Figures 1 and 2, along with the upper and lower bounds $U(\lambda,t,w)$ and $L(\lambda,t,w)$ provided by Proposition 3 and the lower bound $\mathit{LB}_{\mathrm{Br}}(w)$ of [Reference Brown14] for NBUE interarrival times given by (5) – the exponential distribution has the NBUE property. Note that the upper bound $U(\lambda,t,w)$ is not continuous due to the integer parts that appear in its formula. Further, notice that the lower bound $L(\lambda,t,w)$ coincides with the tail distribution of $G_{t}$ (cf. Theorem 3). In the same graphs, we also include the limiting distribution given by Theorem 4.

Figure 1.

The exact tail distribution of the minimal spacing $D_{t}$ with the upper and lower bounds of Proposition 3 and the lower bound of (5). The green solid line corresponds to the limiting exponential distribution of Theorem 4.

Figure 2.

The exact tail distribution of the minimal spacing $D_{t}$ with the upper and lower bounds of Proposition 3 and the lower bound of (5). The limiting exponential distribution (green solid line) approaches the exact one as $\lambda t$ increases.

Figure 2 shows how the limiting distribution (green solid line) approximates the exact distribution of the minimal spacing as $\lambda t$ increases. This approximation seems to be satisfactory even for moderate values of the parameters.

Next, in Figure 3 we illustrate the exact tail distribution of the minimal spacing in the case of a renewal process with Erlang $\mathcal{G}(2,\lambda)$ interarrival times (cf. Proposition 4). The upper bound $\mathit{UB}(w)$ is given by (11) for $a=2$ due to the IFR property of the $\mathcal{G}(2,\lambda)$ distribution. On the other hand, the lower bound $\mathit{LB}(w)$ is the tail distribution of $G_{t}$ since $G_{t}\preceq_{\mathrm{st}}D_{t}$ (cf. Theorem 3). In this figure, we also include the lower bound $\mathit{LB}_{\mathrm{Br}}(w)$ given by (5) since $\mathcal{G}(2,\lambda)$ is NBUE.

Finally, we numerically investigate the case when the interarrival times of a renewal process follow the uniform distribution $\mathcal{U}(0,1)$ . Figure 4 depicts the exact tail distribution of $D_t$ given by Proposition 5(i), along with the approximating formula A(w) of Proposition 5(ii). Here, the tail distribution of $G_{t}$ is also included as the lower bound $\mathit{LB}(w)$ (cf. Theorem 3). We observe that the approximation A(w) is almost equal to the exact formula even for relatively small values of t (e.g. at the right graph, their difference is after the fourth decimal place). In these numerical comparisons, we also include the lower bound $\mathit{LB}_{\mathrm{Br}}(w)$ given by (5) since $\mathcal{U}(0,1)$ is NBUE.

Figure 3.

The exact tail distribution of the minimal spacing $D_{t}$ when the interarrival times have the Erlang $\mathcal{G}(2,\lambda)$ distribution. The upper bound $\mathit{UB}(w)$ , cf. (11), follows from the IFR property of $\mathcal{G}(2,\lambda)$ , while the lower bound $\mathit{LB}(w)$ is a consequence of the stochastic ordering relation of Theorem 3. The lower bound $\mathit{LB}_{\mathrm{Br}}(w)$ is given by (5).

Figure 4.

The exact tail distribution of the minimal spacing $D_{t}$ when the interarrival times have the uniform $\mathcal{U}(0,1)$ distribution. The lower bound $\mathit{LB}(w)$ (red dashed line) follows from the stochastic ordering relation of Theorem 3, while the approximation A(w) (green dashed line) is given by Proposition 5(ii). The lower bound $\mathit{LB}_{\mathrm{Br}}(w)$ is given by (5).

It is worth stressing that all the exact formulae in these figures were also numerically confirmed via Monte Carlo simulation (the respective empirical distribution functions were not included in the graphs in order to keep them simple).

5. Examples of applications

Hypothesis testing

The procedure described in the quality control example could be used more widely to examine whether the interarrival times of a renewal process $\{Y_{t},t\geq0\}$ are i.i.d. from a specific distribution F. For example, we can exploit this simple scheme to test the homogeneity of a Poisson process. In particular, we would like to test the null hypothesis

\begin{align*} H_{0}\colon\{Y_{t},t\geq0\}\text{ is a Poisson counting process with constant intensity }{\lambda}\end{align*}

versus the alternative hypothesis

\begin{align*}H_{1}\colon\text{there exists a subinterval of }[0,t]\text{ where the intensity is higher than }{\lambda}.\end{align*}

It is obvious that, under $H_{1}$ , we may observe events in a certain subinterval more frequently than expected under $H_{0}$ , leading to smaller observed spacings between them. Therefore, the null hypothesis could be rejected when the minimal observed spacing, say $d_{t}$ , is significantly smaller than expected under $H_{0}$ , i.e. when the p-value $=\mathbb{P}(D_{t} < d_{t} \mid H_{0})$ is smaller than the significance level $a\in(0,1)$ of the test.