2. Preliminaries

2.1. The switch process and its expected value

Let $T_k$ , $k=1,2,3,\ldots$ , be a sequence of i.i.d. non-negative random variables with the distribution function F, which is assumed to be absolutely continues with respect to the Lebesgue measure. Additionally, let the corresponding density f associated with F be bounded on any closed interval of the positive half-line. Define a renewal count process for $t\in[0,\infty)$ by

\begin{equation*} N(t)= \begin{cases} \sup \big\{n \in \mathbb N; \sum_{k=1}^n {T_k} \leqslant t \big\}, & t\geqslant T_1, \\[5pt] 0, & 0 \leqslant t<T_1. \end{cases}\end{equation*}

In other words, N(t) is the number of renewal events up to a time point t.

Definition 1. Let N(t), $t \geqslant 0$ , be a count process. Then the switch process is defined by $X(t)=(\!-\!1)^{N(t)}$ , $t\geqslant 0$ .

The process X(t) switches between the values 1 and $-1$ at each renewal event, hence the name.

One of the main objects of interest is the expected value function of the switch process $E(t)=\mathbb{E}X(t)$ . The relation between E and the switching time distribution, F, is implicit in the time domain. There exist some elementary properties of E(t), which are important but straightforward to derive; see, for example, [Reference Cox3]. First, we have the limiting results for $t\geqslant0$ , which follow from the key renewal theorem:

(1) \begin{equation} \lim_{t \rightarrow 0^+} E(t)=1, \qquad \lim_{t \rightarrow \infty} E(t)=0.\end{equation}

The existence of E $^{\prime}$ (t) is of importance for the main results of the paper. Under the assumptions stated in this section and assuming $\sup_{t>0}f(t)<\infty$ on F(t), E $^{\prime}$ (t) exists and is well-defined. The last assumption can be relaxed. Specifically, there needs to exist $l\in \mathbb{N}\colon \sup_{u>0}f^{\ast l}(u)<\infty$ , which allows for more general switching time distributions, e.g. those with unbounded densities at zero.

Let $\mathcal{L}(\!\cdot\!)$ denote the Laplace transform and, in particular, let $\Psi_F(s)=\mathcal{L}(f)(s)$ , where f is the derivative of F when it exists. The Laplace transform of this probability-generating function is well-known and has the following form for $s>0$ :

(2) \begin{equation} \mathcal{L}(E)(s) = \frac{1}{s}\frac{1-\Psi_F(s)}{1+\Psi_F(s)}.\end{equation}

This expression is easily solved for the switching time distribution,

(3) \begin{equation} \Psi_F(s) = \frac{1-s\mathcal{L}(E)(s)}{1+s\mathcal{L}(E)(s)}.\end{equation}

Although the above explicit relations tie the distribution of the switch process and its expected value, they do not provide an explicit condition when a function E is the expected value of a switch process. Naturally, by Bernstein’s theorem, see [Reference Feller5, Theorem 1, p. 415], we could say that this is the case whenever the right-hand side of the above equation is a complete monotone function, but this condition is not easy to check in a concrete case. In Section 3, easy-to-check conditions for E are presented.

3. Switch processes with monotonic expected value function

In this section we fully characterize switch processes with monotonic expected value functions. For the main result, we recall the assumptions on the switching time distribution: F(t) has support on $(0,\infty)$ with a density f(t) for which there exists $l\in \mathbb{N}\colon \sup_{t>0}f^{\ast l}(t)<\infty$ (a technical requirement for the existence of E $^{\prime}$ (t)).

Proof. (i) $\Rightarrow$ (ii): Since $F(t)\in GD(2)$ it has the following Laplace transform, as described in Section 2:

\begin{equation*} \Psi_F(s) = \frac{\frac{1}{2} \Psi_{\tilde{F}(s)}}{1- \frac{1}{2} \Psi_{\tilde{F}(s)}}. \end{equation*}

Substituting this into (2), we have

\begin{equation*} \mathcal{L}(E)(s) = \frac{1}{s} \frac{1-\dfrac{\frac{1}{2} \Psi_{\tilde{F}}(s)}{1-\frac{1}{2}\Psi_{\tilde{F}}(s)}} {1+\dfrac{\frac{1}{2} \Psi_{\tilde{F}}(s)}{1- \frac{1}{2} \Psi_{\tilde{F}}(s)}} = \frac{1}{s} (1- \Psi_{\tilde{F}}(s)), \end{equation*}

which is equivalent to $s\mathcal{L}(E)(s)-1 = -\Psi_{\tilde{F}}(s)$ . The existence of E $^{\prime}$ (t) is needed in order to use the Laplace transform $\mathcal{L}(E')(s)=s\mathcal{L}(E)(s)-E(0)$ . It follows from the stated assumptions by a rather standard although technical argument, see [Reference Daley and Vere-Jones4, Exercise 4.4.3]. Using the above-stated property of the Laplace transform and the limits of E(t), $\mathcal{L}(\!-\!E')(s) = \Psi_{\tilde{F}}(s)$ . By taking the inverse Laplace transform, this implies that $-E'(t)$ is a probability density function. Therefore, to satisfy the limiting results of (1), E(t) must satisfy the conditions of (ii).

(ii) $\Rightarrow$ (i): Under the assumptions of (ii) and the limits of (1) we have

\begin{equation*} \int_0^\infty E'(t)\,{\mathrm{d}} t = \lim_{t \rightarrow \infty} E(t) - \lim_{t \rightarrow 0} E(t)=-1; \end{equation*}

$-E'(t)$ is thus a probability density function. Combining this with the derivative property of the Laplace transform and the limits in (1), (3) becomes

\begin{equation*} \Psi_F (s) = \frac{1-s\mathcal{L}(E)(s)}{1+s\mathcal{L}(E)(s)} = \frac{\mathcal{L}(\!-\!E')(s)}{2-\mathcal{L}(\!-\!E')(s)} = \frac{\frac{1}{2} \mathcal{L}(\!-\!E')(s) }{1-\frac{1}{2} \mathcal{L}(\!-\!E')(s)}. \end{equation*}

This is the Laplace transform of a GD(2) distribution, as described in Section 2. Therefore, $F(t)\in GD(2)$ with the divisor $-E'(t)$ , which yields (i).

Theorem 1 directly relates functional properties of the expected value of the switch process with the switching time distribution for the class of GD(2) distributions. By combining Theorem 1 and properties of E(t) derived in Section 2, a partial solution can be obtained for the case when the switching time distribution belongs to GD(2). To highlight this partial characterization we have the following corollary, which follows from the second part of the proof of Theorem 1.

Corollary 2 gives an explicit representation of the distribution function and density for the 2-geometric divisor of the switching time distribution in terms of E(t).

Corollary 2. Let the switching time distribution, F(t), belong to GD(2), with the divisor $\tilde{F}(t)$ ; then, for $t\geqslant 0$ ,

\begin{equation*} E(t) = 1-\tilde{F}(t), \qquad E'(t) = -\tilde{f}(t). \end{equation*}

Proposition 2 can be used to extend the results of Theorem 1.

However, the opposite is not necessarily true, i.e. a non-negative and decreasing expected value function does not necessarily imply an r-geometric divisible switching time for $r>2$ .

Let us consider a switch process constructed from a count process N(t) and satisfying the conditions of Theorem 1. Further, let $\tilde{N}(t)$ be a count process with the arrival times distributed according to the divisor of this switch process. The two count processes are related through thinning. More specifically, N(t) is a thinning of $\tilde{N}(t)$ , with the probability of thinning equal to $\frac{1}{2}$ . Thus we have the following result.

From a given trajectory of N(t), the trajectory of process $\tilde{N}(t)$ cannot be recovered, in general. However, it follows from Corollary 1 that the distribution of arrival times of $\tilde{N}(t)$ can be recovered. For further relations between geometric divisibility of the switching time distribution and the thinned renewal processes, see [Reference Yannaros11, Reference Yannaros12].

4. The autocovariance of the stationary switch process

A stationary version of the switch process can be constructed by addressing the behavior around zero. Let $\mu<\infty$ be the expected value of the switching time distribution, and $((A,B),\delta )$ be non-negative random variables, mutually independent and independent of X(t), such that $\delta$ takes values $\{-1,1\}$ with equal probability and such that $f_{A,B}(a,b)=({1}/{\mu})f_T(a+b)$ so that the marginals of $f_{A,B}$ are $f_A(t) = f_B(t) = ({1-F_T(t)})/{\mu}$ .

Definition 3. Let $X_+(t)$ and $X_-(t)$ be two independent switch processes, and $((A,B),\delta )$ be as described above. Define

\begin{equation*} Y(t) = \begin{cases} - \delta, & -B < t <A, \\[5pt] \delta X_+(t-A), & t \geqslant A, \\[5pt] -\delta X_-(\!-\!(t+B)), & t \leqslant -B. \end{cases} \end{equation*}

Then Y(t) is called a stationary switch process.

The stationarity of Y(t) follows from standard results in renewal theory. In the same way as the switch process is characterized by its expected value function E(t), the stationary switch process is characterized by its covariance function C(t). There exists a relation between E(t) and C(t) presented in the next theorem.

Theorem 2. Let C(t) be the covariance of the stationary switch process, E(t) be the expected value function of the switch process, and $\mu$ be the expected value of the switching time distribution. Then, for $t\geqslant0$ ,

\begin{equation*} C'(t)=-({2}/{\mu})E(t).\end{equation*}

Proof. Starting with the covariance of Y(t), and utilizing symmetry, we have

\begin{equation*} (\!-\!Y(t) \mid \delta=1) \stackrel{\mathrm{d}}{=}(Y(t) \mid \delta=-1)\end{equation*}

and, for $t>0$ ,

\begin{align*} C(t) = \mathbb{E}(\mathbb{E}(Y(t)Y(0) \mid \delta)) & = -\mathbb{E}(Y(t) \mid \delta=1) \\[5pt] & = -\int_0^\infty \mathbb{E}(Y(t) \mid \delta=1, A=x) f_{A \,\vert\, \delta=1}(x)\,{\mathrm{d}} x \\[5pt] & = -\bigg(\int_0^t\mathbb{E}(\delta X(t-x)\mid\delta=1,A=x)f_A(x)\,{\mathrm{d}} x + \int_t^\infty(\!-\!1)f_A(x)\,{\mathrm{d}} x\bigg). \end{align*}

Since $E(t-x)=0$ , for $x>t$ we obtain $C(t) = 1-F_A(t)-(E\ast f_A)(t)$ . Using the above expression and (2),

\begin{align*} \mathcal{L}(C)(s) = \frac{1}{s} - \mathcal{L}(f_A)(s) \bigg(\frac{1}{s}+\frac{1}{s}\frac{1-\Psi_F(s)}{1+\Psi_F(s)}\bigg) & = \frac{1}{s} - \frac{1-\Psi_F(s)}{\mu s} \bigg(\frac{2}{s}\frac{1}{1+\Psi_F(s)}\bigg) \\[5pt] & = \frac{1}{s}\bigg(1 - \frac{2}{\mu}\mathcal{L}(E)(s)\bigg). \end{align*}

Using $\mathcal{L}(f')(s)=s\mathcal{L}(f)(s)-f(0)$ and $C(0)=1$ ,

\begin{equation*} s\mathcal{L}(C)(s)-1 = -\frac{2}{\mu}\mathcal{L}(E)(s), \qquad C'(t) = -\frac{2}{\mu}E(t). \end{equation*}

Theorem 2 allows us to use functional properties of the expected value of the switch process to investigate the covariance of the stationary switch process. In particular, combining Theorem 2 with Theorem 1 and Proposition 1 yields a partial characterization of the covariance functions.

By combining Theorem 1 and Theorem 2, the divisor’s density and distribution can be derived from the covariance function.

Corollary 6. Let C(t) be the covariance of the stationary switch process and the switching time distribution belong to GD(2), with the divisor distribution $\tilde{F}$ ; then, for $t\geqslant0$ ,

\begin{equation*} 1+\frac{\mu}{2} C'(t) = \tilde{F}(t), \qquad \frac{\mu}{2}C''(t) = \tilde{f}(t), \end{equation*}

where $\mu=-2C'(0^+)$ .

The identification of $\mu$ does not require geometric divisibility, since it follows from the limits of E(t) and Theorem 2.

Even if the switching time distribution does not belong to GD(2) Theorem 2 is still applicable, as illustrated in the next example.