Local properties for $1$-dimensional critical branching Lévy process
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912072045953024 |
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| author | Hou, Haojie Ren, Yan-Xia Song, Renming |
| author_facet | Hou, Haojie Ren, Yan-Xia Song, Renming |
| contents | Consider a one dimensional critical branching Lévy process $((Z_t)_{t\geq 0}, \mathbb {P}_x)$. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction to some $α$-stable distribution with $α\in (1, 2)$, and that the underlying Lévy process $(ξ_t)_{t\geq 0}$ is non-lattice and has finite $2+δ^*$ moment for some $δ^*>0$. We first prove that $$t^{\frac{1}{α-1}}\left(1- \mathbb{E}_{\sqrt{t}y}\left(\exp\left\{-\frac{1}{t^{\frac{1}{α-1}-\frac{1}{2}}}\int h(x) Z_t(\mathrm{d}x) -\frac{1}{t^{\frac{1}{α-1}}} \int g\left(\frac{x}{\sqrt{t}}\right)Z_t(\mathrm{d}x)\right\}\right)\right)$$
converges as $t\to\infty$ for any non-negative bounded Lipschtitz function $g$ and any non-negative directly Riemann integrable function $h$ of compact support.
Then for any $y\in \R$ and bounded Borel set of positive Lebesgue measure with its boundary having zero Lebesgue measure, under a higher moment condition on $ξ$, we find the decay rate of the probability $\mathbb {P}_{\sqrt{t}y}(Z_t(A)>0)$. As an application, we prove some convergence results for $Z_t$ under the conditional law $\mathbb {P}_{\sqrt{t}y}(\cdot| Z_t(A)>0).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local properties for $1$-dimensional critical branching Lévy process Hou, Haojie Ren, Yan-Xia Song, Renming Probability Consider a one dimensional critical branching Lévy process $((Z_t)_{t\geq 0}, \mathbb {P}_x)$. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction to some $α$-stable distribution with $α\in (1, 2)$, and that the underlying Lévy process $(ξ_t)_{t\geq 0}$ is non-lattice and has finite $2+δ^*$ moment for some $δ^*>0$. We first prove that $$t^{\frac{1}{α-1}}\left(1- \mathbb{E}_{\sqrt{t}y}\left(\exp\left\{-\frac{1}{t^{\frac{1}{α-1}-\frac{1}{2}}}\int h(x) Z_t(\mathrm{d}x) -\frac{1}{t^{\frac{1}{α-1}}} \int g\left(\frac{x}{\sqrt{t}}\right)Z_t(\mathrm{d}x)\right\}\right)\right)$$ converges as $t\to\infty$ for any non-negative bounded Lipschtitz function $g$ and any non-negative directly Riemann integrable function $h$ of compact support. Then for any $y\in \R$ and bounded Borel set of positive Lebesgue measure with its boundary having zero Lebesgue measure, under a higher moment condition on $ξ$, we find the decay rate of the probability $\mathbb {P}_{\sqrt{t}y}(Z_t(A)>0)$. As an application, we prove some convergence results for $Z_t$ under the conditional law $\mathbb {P}_{\sqrt{t}y}(\cdot| Z_t(A)>0).$ |
| title | Local properties for $1$-dimensional critical branching Lévy process |
| topic | Probability |
| url | https://arxiv.org/abs/2410.10066 |