Jump-over functionals for integer-valued Poisson processes. (Ukrainian, English) Zbl 1050.60051

Gusak, D. V.

Teor. Jmovirn. Mat. Stat. 68, 24-36 (2003); translation in Theory Probab. Math. Stat. 68, 27-40 (2004).

Let \(\xi(t),\;\xi(0)=0\), \(t\geq0\), be integer-valued Poisson process. Let us define functionals \(\tau^{+}(x)=\inf\{t: \xi(t)>x\}\), \(\overline\tau^{+}(x)=\inf\{t: \xi(t)\geq x\}\), \(\gamma^{+}(x)=\xi^{+}(\tau^{+}(x))-x\), \(\overline\gamma^{+}(x)= \xi^{+}(\overline\tau^{+}(x))-x\), \(\gamma_{+}(x)=x-\xi^{+}(\tau^{+}(x)-0)\), \(\overline\gamma_{+}(x)= x-\xi^{+}(\overline\tau^{+}(x)-0)\), \(\gamma^{+}_{x}= \gamma^{+}(x)+\gamma_{+}(x)\), \(\overline \gamma^{+}_{x}= \overline\gamma^{+}(x)+ \overline\gamma_{+}(x)\), where \(\xi^{+}(t)=\sup_{0\leq u\leq t}\xi(u)\). The author derives joint distribution of \((\tau^{+}(x),\gamma^{+}(x))\) and \((\overline\tau^{+}(x), \overline\gamma^{+}(x))\). The distributions for the case of semi-continuous processes are obtained in more precise form and limit relationships are derived as \(x=0\) and \(x\to\infty\).

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60G55

Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

jump-over functionals; integer-valued Poisson processes; joint distribution