-ASYMPTOTIC OF EXTREMES OF MOVING MINIMA IN -ARRAYS OF INDEPENDENT RANDOM VARIABLES
Jadwiga Dudkiewicz · Demonstratio Mathematica · 1996
IntroductionLet {X nt i,i = 1,..., n, n = 1,2,...} be an array of independent random variables, which have identical distribution function F n for fixed n.We define sequence of maxima of moving minima based on array {X n> i} (!)M n,l = .ma*j.. .min X n,i, where 1 i,..., X n>n with identical distribution function F n .The system will fail if and only if a least m consecutive components fail.The lifetime of system is therefore random variable Mn,m defined by (1).Consecutive-m-out-of-n systems have extensive applications.Recently they have been proposed to model telecommunication systems and oil pipelines, vacuum in accelerators, computer ring network and spacecraft relay station (see e.g.[1], [3], [4] and papers referred there).Many authors have been interested in the problem of investigation of asymptotic lifetime of Mn]m system (see e.g.[3], [4]).Recently, E.R. Canfield and W.P. McCormick have studied the asymptotic of Mn)n in the case, where both n and m = m n change (see [1]).Among other things they showed, that if TfX (2) ----»• 0, as n -> oo In n then (3) P{Mi% < u n } e~e x , as n ^ oo