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Zbl 0987.60028
Mallor, F.; Omey, E.
Shocks, runs and random sums. (English)
[J] J. Appl. Probab. 38, No.2, 438-448 (2001). ISSN 0021-9002

Let $(A,B)$, $(A(i),B(i))$, $i= 1,2,\dots$, be i.i.d. nonnegative random vectors and let $S(n)= B(1)+\cdots+ B(n)$, $N(k)= \min\{j: A(j-i)\in R,i= 0,\dots, k-1\}$, $Y(k)= S(N(k))$ and $M(k)= \max\{A(i):1\le i\le N(k)\}$, for a fixed subset $R$ of $(0,\infty)$. 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.\par 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 $P(A\in R)$, and its variance are derived. A condition on $1- E\exp(-sB)$ as $s\to 0$ ensures the asymptotic behaviour in distribution of $Y(k)$ as $k\to\infty$ for fixed $P(A\in R)$. A similar one is derived for $k$ fixed and $R$ small such that $P(A\in R)\to 0$. The conditions imply a form of regular variation. These derivations use only the recurrence for $E\exp(- sY)$.
[A.J.Stam (Winsum)]

MSC 2000:: *60E10 Transforms of probability distributions
60F05 Weak limit theorems
60K10 Appl. of renewal theory
90B25 Reliability, etc.
26A12 Rate of growth of functions of one real variable
60G50 Sums of independent random variables

Keywords: random sums; reliability; insurance; asymptotics

Cited in: Zbl 1122.60075 Zbl 1005.60048

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