zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Shocks, runs and random sums. (English) Zbl 0987.60028
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. 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 $E\exp(- sY)$.

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
Full Text: DOI