On renewal theory for cluster processes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866914803438583808 |
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| author | Basrak, Bojan Dajaković, Marina |
| author_facet | Basrak, Bojan Dajaković, Marina |
| contents | We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $(ξ_i,X_i)_{i \geq 0}$, where $ξ_0$ denotes a finite point process on $\mathbb{R}$ and $X_0$ denotes a nonnegative random variable of finite mean, we consider the renewal sequence $T_i = X_0+\cdots + X_i$, $i \geq 0$, and corresponding renewal cluster process $ ξ(\cdot )=\sum_{i\geq0}ξ_i(\,\cdot -T_i)$. Under mild assumptions on the distribution of $(ξ,X)$, we show by coupling methods that the generalized versions of Blackwell's renewal theorem, key renewal theorem, extended renewal theorem and elementary renewal theorem still hold, even with dependence between $ξ_i$'s and $X_i$'s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03749 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On renewal theory for cluster processes Basrak, Bojan Dajaković, Marina Probability 60K05, 60G55, 60B10 We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $(ξ_i,X_i)_{i \geq 0}$, where $ξ_0$ denotes a finite point process on $\mathbb{R}$ and $X_0$ denotes a nonnegative random variable of finite mean, we consider the renewal sequence $T_i = X_0+\cdots + X_i$, $i \geq 0$, and corresponding renewal cluster process $ ξ(\cdot )=\sum_{i\geq0}ξ_i(\,\cdot -T_i)$. Under mild assumptions on the distribution of $(ξ,X)$, we show by coupling methods that the generalized versions of Blackwell's renewal theorem, key renewal theorem, extended renewal theorem and elementary renewal theorem still hold, even with dependence between $ξ_i$'s and $X_i$'s. |
| title | On renewal theory for cluster processes |
| topic | Probability 60K05, 60G55, 60B10 |
| url | https://arxiv.org/abs/2211.03749 |