Abstract
A compound Poisson process is of the form \(S_{N_\lambda } = \sum\nolimits_{j = 1}^{N_\lambda } {Z_j }\) where Z, Z 1, Z 2, are arbitrary i.i.d. random variables and N λ is an independent Poisson random variable with parameter λ. This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate \(P(S_{N_\lambda } \geqslant \lambda a)\). The truncation level introduced depends only on λ and Z and not on the overall exceedance level λa.
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Hahn, M.G., Klass, M.J. Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes. Journal of Theoretical Probability 11, 535–559 (1998). https://doi.org/10.1023/A:1022696125272
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DOI: https://doi.org/10.1023/A:1022696125272