Abstract
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 rth 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.
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The datasets/materials used and/or analysed during the current study are available from the authors on reasonable request.
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All authors contributed to the study conception and design. Material preparation and analysis were performed by S. Chadjiconstantinidis, M.V. Koutras and F.S. Milienos. The first draft of the manuscript was written by S. Chadjiconstantinidis. All authors read and approved the final manuscript.
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Chadjiconstantinidis, S., Koutras, M.V. & Milienos, F.S. The distribution of extended discrete random sums and its application to waiting time distributions. Methodol Comput Appl Probab 25, 49 (2023). https://doi.org/10.1007/s11009-023-10027-0
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DOI: https://doi.org/10.1007/s11009-023-10027-0
Keywords
- Runs
- Multiple run occurrences
- Probability generating functions
- Recursive schemes
- Markov chain imbedding technique
- Collective risk model
- Claims distribution