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

Kushnir, O. O.

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 \(\Psi_{\varepsilon}(s,x)= P\{ \Theta_{\varepsilon}(s)<x\}\), \(p(\varepsilon)= {1\over EF}\int_0^\varepsilon (1-F(x))(1-G(x)) dx\).

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

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

Keywords:

independent random variables; renewal process; limit theorem; Laplace transform; \(\varepsilon\)-approach