The distribution of extended discrete random sums and its application to waiting time distributions. (English) Zbl 1518.60015
Summary: In this work, we derive the exact distribution of a random sum of the form \(S=U+X_1+\ldots +X_M\), where the \(X_j\)’s are independent and identically distributed positive integer-valued random variables, independent of the non-negative integer-valued random variables \(M\) and \(U\) (which are also independent). Efficient recurrence relations are established for the probability mass function, cumulative distribution function and survival function of \(S\) as well as for the respective factorial moments of it. These results are exploited for deriving new recursive schemes for the distribution of the waiting time for the \(r\)th appearance of run of length \(k\), under the non-overlapping, at least and overlapping scheme, defined on a sequence of identically distributed binary trials which are either independent or exhibit a \(k\)-step dependence.
MSC:
| 60E05 | Probability distributions: general theory |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 62P25 | Applications of statistics to social sciences |
| 62P30 | Applications of statistics in engineering and industry; control charts |
Keywords:
runs; multiple run occurrences; probability generating functions; recursive schemes; Markov chain imbedding technique; collective risk model; claims distributionReferences:
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