Exact Asymptotics for the Distribution of the Time of Attaining the Maximum for a Trajectory of a Compound Poisson Process with Linear Drift

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...

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Bibliographic Details
Published in:Siberian advances in mathematics 2020, Vol.30 (1), p.26-42
Main Author: Mosyagin, V. E.
Format: Article
Language:English
Subjects:
Asymptotic properties
Distribution functions
Mathematics
Mathematics and Statistics
Poisson density functions
Random processes
Statistical analysis
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Summary: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.
ISSN:1055-1344
1934-8126
DOI:10.3103/S1055134420010034