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A study of a highly reliable system with protection by the Rényi theorem. (English. Ukrainian original) Zbl 0923.60092
Theory Probab. Math. Stat. 55, 121-128 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 117-124 (1996).
Let $$(\xi_n)$$, $$(\eta_n)$$, $$(\zeta_n)$$, $$n \geq 1$$, be three sequences of independent and positive random variables with distribution functions $$F$$, $$G$$ and $$J$$, respectively. Let $$T_0 =0$$, $$S_0=-s$$ $$(s>0)$$, $$T_n' =T_{n-1} +\zeta_n$$, $$T_n =T_n' +\eta_n$$, $$S_n = S_{n-1}+\zeta_n$$, $$n \geq 1.$$ Let $$\tau_s$$ be the moment of the first hit of $$S_n$$ in the random segment $$[T_n, T_n'].$$ The asymptotic behaviour of the distribution function $$\varphi_s(x) = P\{ \tau_s <x\}$$ is investigated. The author proves two theorems.
Theorem 1. Let the following assumptions hold: (a) $$F$$ has not a discrete component; (b) for some $$i \in N$$ the function $$F^{\ast i}$$ has absolutely continuous component; (c) $$E F < +\infty$$, $$E G < +\infty$$, $$F(+0) =0$$, $$G(+0)=0$$. Then $\lim_{EJ \to 0} \sup_{t \geq 0} \sup_{s>0} \biggl | \varphi_s(t) -1 + \exp \biggl\{ -{tp \over E G} \biggr\} \biggr | =0.$ Theorem 2 contains assumptions under which the following assertion holds true $\lim_{n \to \infty} \sup_{t \geq 0} \sup_{s>0}\biggl | \varphi_{s,n}(t) -1 + \exp \biggl\{ -{tp_n \over E G_n} \biggr\} \biggr | =0.$

MSC:
 60K10 Applications of renewal theory (reliability, demand theory, etc.) 90B25 Reliability, availability, maintenance, inspection in operations research