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

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 $$x\to\infty$$.

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)