Abstract
We consider the random process at − v+(pt) + v−(−qt), t ∈ (−∞, −), where v− and v+ are independent standard Poisson processes if t ≥ 0 and v−(t) = v+(t) = 0 if t < 0. Under certain conditions on the parameters a, p, and q, we study the distribution function G = G(x) of the time of attaining the maximum for a trajectory of this process. In the present article, we find an exact asymptotics for the tails of G. We also find a connection between this problem and the statistical problem of estimation of an unknown discontinuity point of a density function.
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Acknowledgements
The author would like to express his deep gratitude to Professor Igor S. Borisov for his suggestion to study asymptotical properties of the distribution function G. The author is also grateful to the anonymous referee for her/his useful comments.
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Russian Text © The Author(s), 2019, published in Matematicheskie Trudy, 2019, Vol. 22, No. 2, pp. 134–156.
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Mosyagin, V.E. Exact Asymptotics for the Distribution of the Time of Attaining the Maximum for a Trajectory of a Compound Poisson Process with Linear Drift. Sib. Adv. Math. 30, 26–42 (2020). https://doi.org/10.3103/S1055134420010034
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DOI: https://doi.org/10.3103/S1055134420010034