标题： 振荡布朗运动中的自组织临界性水平：MLE 的$n$-一致性与稳定的泊松型收敛 显示英文标题

标题： The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE

摘要： 对于具有自组织临界性水平的离散观测振荡布朗运动路径$\rho_0$，我们在内填充渐近框架下证明了最大似然估计（MLE）是$n$-相合的，其中$n$表示样本量，并推导出其相对于稳定收敛的极限分布。 由于这个齐次马尔可夫过程的转移密度甚至在$\rho_0$中也不连续，因此分析高度非标准。 因此，出现了有趣且有些意想不到的现象：似然函数分解为多个分量，每个分量根据参数$\rho$接近$\rho_0$的程度贡献非常不同。 相应地，最大似然估计（MLE）依次被排除在一个紧集之外，一个$1/\sqrt{n}$- 邻域之外，最后是一个$1/n$- 邻域之外，渐近地接近$\rho_0$。 得到稳定收敛的关键论据是利用顺序适当地重新标度的局部对数似然函数（作为时间过程）的半鞅结构。 无论是顺序上还是作为$\rho$中的过程，它在稳定极限下都表现出双变量泊松行为，其强度是局部时间在$\rho_0$处的倍数。 显示英文摘要

摘要： For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $\rho_0$, we prove in the infill asymptotics that the MLE is $n$-consistent, where $n$ denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in $\rho_0$, the analysis is highly non-standard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument $\rho$ is to $\rho_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neigborhood of $\rho_0$ asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in $\rho$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $\rho_0$.