Some Results and Applications of Geometric Counting Processes

Abstract

Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.

Appendix: Proof of Proposition 2.2

Fix $\textbf {t}=(t_{1},t_{2},\ldots ,t_{m}) \in \mathcal {T}^{+}_{m}$. From Eq. 7 we have

$$ f_{\mathbf{T_{m}}}(\textbf{t}) = (-1)^{m} \sum\limits_{(k_{1},k_{2}, \ldots,k_{m}) \in \mathcal{A}} \frac{\partial^{m} } {{\partial t_{1} \partial t_{2} \cdots \partial t_{m}}} \ p_{(k_{1},k_{2}, \ldots,k_{m})}(0,t_{1},t_{2}, \ldots,t_{m}). $$

(38)

Recall now that, for $\textbf {k}=(k_{1},k_{2}, \ldots ,k_{m}) \in \mathcal {A}$,

$$p_{\textbf{k}}(0,t_{1},t_{2}, \ldots,t_{m}) = {{\sum}_{i = 1}^{m} k_{i} \choose k_{1}, k_{2}, \ldots,k_{m}} \frac{\lambda^{\sum k_{i}}}{[1+\lambda t_{m}]^{1+\sum k_{i}}} \ \big[t_{1}^{k_{1}} (t_{2}-t_{1})^{k_{2}} {\cdots} (t_{m}-t_{m-1})^{k_{m}} \big], $$

and observe that it holds

$$ \frac{\partial^{m-1}}{{\partial t_{1} \partial t_{2} {\cdots} \partial t_{m-1}}} \big[t_{1}^{k_{1}} (t_{2}-t_{1})^{k_{2}} {\cdots} (t_{m}-t_{m-1})^{k_{m}} \big] \ne 0 $$

(39)

if and only if the term t₁t₂⋯t_m appears in expansion of the product $t_{1}^{k_{1}} (t_{2}-t_{1})^{k_{2}} \cdots (t_{m}-t_{m-1})^{k_{m}}$. With a straightforward computation of such product, and considering the constraints in Eq. 8, it is easy to see that the condition

$$ \frac{\partial^{m} }{{\partial t_{1} \partial t_{2} {\cdots} \partial t_{m}}} \ p_{\textbf{k}} (0,t_{1},t_{2},\ldots,t_{m})\neq 0 $$

(40)

is fulfilled if and only if k₁ + k₂ + … + k_m = m − 1. Recalling that it should be k₁ = 0, and again considering the constrains (8), we have that Eq. 40 holds if, and only if, k₁ = 0 and k_i = 1 for all i = 2, 3,…,m. In this case we have

$$\begin{array}{@{}rcl@{}} & & \frac{\partial^{m-1} }{\partial t_{1} \partial t_{2} {\cdots} \partial t_{m-1}} \ p_{(0,1,1, \ldots,1)}(0,t_{1},t_{2}, \ldots,t_{m}) \\ &=& \frac{\partial^{m-1}}{\partial t_{1}\partial t_{2}\ldots\partial t_{m-1}} \ {m-1 \choose 0, 1, \ldots,1} \ \lambda^{m-1} \frac{{\prod}_{i = 2}^{m} (t_{i}-t_{i-1})}{[1+\lambda t_{m}]^{m}} \\ &=& (m-1)! \ \lambda^{m-1} \ \frac{\partial^{m} }{\partial t_{1}\partial t_{2}\ldots\partial t_{m}} \ \frac{{\prod}_{i = 2}^{m} (t_{i}-t_{i-1})}{[1+\lambda t_{m}]^{m}} \\ &=& \frac{(m-1)! \ \lambda^{m-1} }{[1+\lambda t_{m}]^{m}}. \end{array} $$

(41)

In conclusion, from Eqs. 38, 39 and 41, for 0 < t₁ < t₂ < … < t_m we have

$$\begin{array}{@{}rcl@{}} f_{\mathbf{T_{m}}}(\textbf{t}) & = & (-1)^{m} \frac{\partial^{m} }{{\partial t_{1} \partial t_{2} {\cdots} \partial t_{m}}} \ p_{(0,1, \ldots,1)}(0,t_{1},t_{2}, \ldots,t_{m}) \\ & = & (-1)^{m} \frac{\partial }{{\partial t_{m}}} \Big[ \frac{\partial^{m-1} }{{\partial t_{1} \partial t_{2} {\cdots} \partial t_{m-1}}} \ p_{(0,1, \ldots,1)}(0,t_{1},t_{2}, \ldots,t_{m}) \Big] \\ & = & (-1)^{m} \frac{\partial }{{\partial t_{m}}} \ \frac{(m-1)! \ \lambda^{m-1} }{[1+\lambda t_{m}]^{m}} \\ & = & \frac{m! \ \lambda^{m} }{[1+\lambda t_{m}]^{m + 1}}, \end{array} $$

(42)

while the density is zero whenever $\textbf {t} \not \in \mathcal {T}^{+}_{m}$. Finally, this gives (9).