## Estimates and convergence for the probability of $$\varepsilon$$-approach of two independent renewal processes.(English. Ukrainian original)Zbl 0955.60081

Theory Probab. Math. Stat. 60, 125-130 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 112-117 (1999).
The author considers two flows $$S_{n}$$ and $$T_{n}$$, $$S_{n}= S_{n-1}+\xi_{n}$$, $$n \geq 1$$, $$S_0 =-s$$ $$(s\geq 0)$$, $$T_{m}=T_{m-1}+\eta_{m}$$, $$m\geq 1$$, $$T_0=0$$, where $$\xi_{n}$$, $$n\geq 1$$, and $$\eta_{n}$$, $$n\geq 1$$, are independent random variables such that $$\xi_{n}\geq 0$$ and $$\eta_{n}\geq 0$$, $$n\geq 1.$$ Let $$P\{ \xi_{n}<t\} =F(t)$$, $$n\geq 1$$, $$P\{ \eta_{n} <t\} =G(t)$$, $$F(0)=G(0)=0$$, $$F(+0)<1$$, $$G(+0)<1.$$ Let $$\Theta_{\varepsilon}(s)$$ be a moment of the first $$\varepsilon$$-approach of the flows $$S_{n}$$ and $$T_{n},$$ that is $\Theta_{\varepsilon}(s)=\inf_{k\geq 1} \Biggl\{ S_{k}: S_{k} \in \bigcup_{n=0}^\infty (T_{n}, T_{n}+ \varepsilon)\Biggr\}.$ The author proves limit theorems for $$\Theta_{\varepsilon}(s)$$ as $$\varepsilon \to 0.$$ For example, he proves the following theorem: Let the following conditions be satisfied: 1. $$F$$ is a distribution function of absolutely continuous type; 2. $$F$$ is a continuous function; 3. $$0< EF<+\infty$$, $$0< EG<+\infty$$. Then the following assertion holds true $\lim_{\varepsilon \to 0} \sup_{s\geq 0} \sup_{x\geq 0}\bigl|\Psi_{\varepsilon}(s,x)-1+ \exp\bigl\{ -p(\varepsilon)x/EG\bigr\}\bigl|=0,$ where $$\Psi_{\varepsilon}(s,x)= P\{ \Theta_{\varepsilon}(s)<x\}$$, $$p(\varepsilon)= {1\over EF}\int_0^\varepsilon (1-F(x))(1-G(x)) dx$$.

### MSC:

 60K05 Renewal theory 60K10 Applications of renewal theory (reliability, demand theory, etc.)