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Shocks, runs and random sums. (English) Zbl 0987.60028

Let (A,B), (A(i),B(i)), i=1,2,, be i.i.d. nonnegative random vectors and let S(n)=B(1)++B(n), N(k)=min{j:A(j-i)R,i=0,,k-1}, Y(k)=S(N(k)) and M(k)=max{A(i):1iN(k)}, for a fixed subset R of (0,). Interpretation: a system subject to load cycles or shocks with magnitudes A(i) and durations or intershock times B(i). Then N(k) is the number of the shock where for the first time k successive shocks have magnitudes in a critical region R. The total duration or time up to this shock is Y(k). Another interpretation in insurance: claims and interclaim times.

The paper derives the Laplace-Stieltjes transform of Y(k), the probability generating function of N(k) and the distribution function of M(k) by means of recurrence w.r. to k. The first moment of Y(k), in terms of EB and P(AR), and its variance are derived. A condition on 1-Eexp(-sB) as s0 ensures the asymptotic behaviour in distribution of Y(k) as k for fixed P(AR). A similar one is derived for k fixed and R small such that P(AR)0. The conditions imply a form of regular variation. These derivations use only the recurrence for Eexp(-sY).


MSC:
60E10Transforms of probability distributions
60F05Central limit and other weak theorems
60K10Applications of renewal theory
90B25Reliability, availability, maintenance, inspection, etc. (optimization)
26A12Rate of growth of functions of one real variable, orders of infinity, slowly varying functions
60G50Sums of independent random variables; random walks