## A logarithmic lower estimate for the mathematical expectation of the first failure moment of a higher-reliable system with protection in a nonstationary regime.(Ukrainian, English)Zbl 1026.60101

Teor. Jmovirn. Mat. Stat. 65, 104-109 (2001); translation in Theory Probab. Math. Stat. 65, 115-122 (2002).
Summary: Let $$(\xi_n)$$, $$(\eta_n)$$, $$(\zeta_n)$$, $$n \geq 1$$, be three sequences of independent and nonnegative random variables with distribution functions $$F$$, $$G$$ and $$J$$, respectively. Let $$T_0 =0$$, $$S_0=-s$$ $$(s>0)$$, $$T_n' =T_n +\zeta_n$$, $$T_{n+1} =T_n' +\eta_n$$, $$S_n=S_{n-1}+\xi_n$$, $$n \geq 1.$$ Let $$\tau_s$$ be the moment of the first hit of the sequence $$(S_k)_{k\geq 1}$$ in the random segment $$[T_n, T_n'],n\geq 1$$, i.e. $$\tau=\inf_{k\geq 1}\left\{S_k: S_k\in\bigcup_{n=1}^{\infty} [T_n, T_n']\right\}.$$ Denote by $$b_s=E\{\tau/S_0=-s\}$$ the conditional expectation of the random variable $$\tau$$. Under some rather strong assumptions on the distribution functions $$F$$, $$G$$ and $$J$$ (existence of exponential moments, in particular) the author finds estimates from below for the conditional expectation $$b_s$$. For more details see the articles by the author [Theory Probab. Math. Stat. 55, 121-128 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 117-124 (1996; Zbl 0923.60092) and ibid. 61, 95-100 (2000) resp. ibid. 61, 91-96 (2000; Zbl 0985.60079)].

### MSC:

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

### Citations:

Zbl 0923.60092; Zbl 0985.60079