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

Let $\left(A,B\right)$, $\left(A\left(i\right),B\left(i\right)\right)$, $i=1,2,\cdots$, be i.i.d. nonnegative random vectors and let $S\left(n\right)=B\left(1\right)+\cdots +B\left(n\right)$, $N\left(k\right)=min\left\{j:A\left(j-i\right)\in R,i=0,\cdots ,k-1\right\}$, $Y\left(k\right)=S\left(N\left(k\right)\right)$ and $M\left(k\right)=max\left\{A\left(i\right):1\le i\le N\left(k\right)\right\}$, for a fixed subset $R$ of $\left(0,\infty \right)$. Interpretation: a system subject to load cycles or shocks with magnitudes $A\left(i\right)$ and durations or intershock times $B\left(i\right)$. Then $N\left(k\right)$ 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 $Y\left(k\right)$. Another interpretation in insurance: claims and interclaim times.

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

##### MSC:
 60E10 Transforms of probability distributions 60F05 Central limit and other weak theorems 60K10 Applications of renewal theory 90B25 Reliability, availability, maintenance, inspection, etc. (optimization) 26A12 Rate of growth of functions of one real variable, orders of infinity, slowly varying functions 60G50 Sums of independent random variables; random walks
##### Keywords:
random sums; reliability; insurance; asymptotics