On certain integral functionals of integer-valued subordinators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908556392923136 |
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| author | Hu, Dongdong Sayit, Hasanjan Xia, Weixuan |
| author_facet | Hu, Dongdong Sayit, Hasanjan Xia, Weixuan |
| contents | It is known that the exponential functional of a Poisson process admits a probability density function in the form of an infinite series. In this paper, we obtain an explicit expression for the density function of the exponential functional of any integer-valued subordinator, and by extension, limit representations for that of an arbitrary pure-jump subordinator. With an added positive drift, the density function is expressed via piecewise basis functions governed by a functional relation. Closed-form density functions for these cases have been established only for a few special instances of Lévy processes in the past literature. Our work substantially advances this line of research by providing an analytical perspective on the distribution of a broad class of exponential Lévy functionals, also suggesting potential methodological extensions to general purely discontinuous Lévy processes. Moreover, we consider arbitrary decreasing functionals of integer-valued subordinators by deriving sufficient and necessary conditions for their convergence, which are then applied to obtain limit-series representations for the density functions of inverse-power functionals. The numerical performance of the proposed formulae is demonstrated through various examples of well-known distributions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On certain integral functionals of integer-valued subordinators Hu, Dongdong Sayit, Hasanjan Xia, Weixuan Probability 60G51, 60E05 It is known that the exponential functional of a Poisson process admits a probability density function in the form of an infinite series. In this paper, we obtain an explicit expression for the density function of the exponential functional of any integer-valued subordinator, and by extension, limit representations for that of an arbitrary pure-jump subordinator. With an added positive drift, the density function is expressed via piecewise basis functions governed by a functional relation. Closed-form density functions for these cases have been established only for a few special instances of Lévy processes in the past literature. Our work substantially advances this line of research by providing an analytical perspective on the distribution of a broad class of exponential Lévy functionals, also suggesting potential methodological extensions to general purely discontinuous Lévy processes. Moreover, we consider arbitrary decreasing functionals of integer-valued subordinators by deriving sufficient and necessary conditions for their convergence, which are then applied to obtain limit-series representations for the density functions of inverse-power functionals. The numerical performance of the proposed formulae is demonstrated through various examples of well-known distributions. |
| title | On certain integral functionals of integer-valued subordinators |
| topic | Probability 60G51, 60E05 |
| url | https://arxiv.org/abs/2509.17199 |