薄过程邻域计数稀疏定律
A law of thin processes with neighbour-count thinning
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中文总结 AI 辅助
研究独立同分布简单点过程叠加后经依赖邻域计数的稀疏化，在临界尺度下极限为非齐次泊松过程，强度经非线性修正。
中文摘要 AI 辅助
我们考虑$\mathbb{R}^d$上$n$个独立同分布简单点过程的叠加$S_n$，并施加一个依赖稀疏化$T_n$，其中点的保留概率取决于其在半径$r_n$内的局部邻域计数。虽然独立稀疏化下的叠加收敛到泊松过程，但我们证明在临界几何尺度$n v_d r_n^d \to \tau \in (0,\infty)$下，局部相互作用在极限中转化为具有修正强度的非齐次泊松点过程。我们证明稀疏化序列$T_n(S_n)$收敛到具有非线性修正强度$\tilde\lambda(x) = \lambda(x)\alpha(\tau\lambda(x))$的泊松过程。
英文摘要
We consider the superposition $S_n$ of $n$ i.i.d. simple point processes on $\mathbb{R}^d$ and apply a dependent thinning $T_n$, where the retention probability of a point depends on its local neighbour count within a radius $r_n$. While superpositions under independent thinning converge to Poisson processes, we show that under a critical geometric scaling $n v_d r_n^d \to τ\in (0,\infty)$, the local interactions are transformed in the limit into an inhomogeneous Poisson point process with modified intensity. We prove that the thinned sequence $T_n(S_n)$ converges to a Poisson process with a non-linearly modified intensity $\tildeλ(x) = λ(x)α(τλ(x))$.
发表机构
- School of Mathematics, Jilin University（吉林大学数学学院）
- Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）
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