Thursday, January 29, 2026 - 16:00 in V3-201
The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE
A talk in the Oberseminar Probability Theory and Mathematical Statistics series by
Angelika Rohde from Universität Freiburg
| Abstract: |
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 $\rho_0$.
Within the CRC this talk is associated to the project(s): B10 |
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