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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 \[ \lim_{n \to \infty} \sup_{t \geq 0} \sup_{s>0}\biggl | \varphi_{s,n}(t) -1 + \exp \biggl\{ -{tp_n \over E G_n} \biggr\} \biggr | =0. \]

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