Renewal Processes Represented as Doubly Stochastic Poisson Processes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917788195487744 |
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| author | Du, Xinlong Honnappa, Harsha |
| author_facet | Du, Xinlong Honnappa, Harsha |
| contents | This paper gives an elementary proof for the following theorem: a renewal process can be represented by a doubly-stochastic Poisson process (DSPP) if and only if the Laplace-Stieltjes transform of the inter-arrival times is of the following form: $$ϕ(θ)=λ\left[λ+θ+k\int_0^\infty\left(1-e^{-θz}\right)\,dG(z)\right]^{-1},$$ for some positive real numbers $λ, k$, and some distribution function $G$ with $G(\infty)=1$. The intensity process $Λ(t)$ of the corresponding DSPP jumps between $λ$ and $0$, with the time spent at $λ$ being independent random variables that are exponentially distributed with mean $1/k$, and the time spent at $0$ being independent random variables with distribution function $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Renewal Processes Represented as Doubly Stochastic Poisson Processes Du, Xinlong Honnappa, Harsha Probability Statistics Theory This paper gives an elementary proof for the following theorem: a renewal process can be represented by a doubly-stochastic Poisson process (DSPP) if and only if the Laplace-Stieltjes transform of the inter-arrival times is of the following form: $$ϕ(θ)=λ\left[λ+θ+k\int_0^\infty\left(1-e^{-θz}\right)\,dG(z)\right]^{-1},$$ for some positive real numbers $λ, k$, and some distribution function $G$ with $G(\infty)=1$. The intensity process $Λ(t)$ of the corresponding DSPP jumps between $λ$ and $0$, with the time spent at $λ$ being independent random variables that are exponentially distributed with mean $1/k$, and the time spent at $0$ being independent random variables with distribution function $G$. |
| title | Renewal Processes Represented as Doubly Stochastic Poisson Processes |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2409.18362 |