The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912960651198464 |
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| author | Brutsche, Johannes Rohde, Angelika |
| author_facet | Brutsche, Johannes Rohde, Angelika |
| contents | For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $ρ_0$, 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 $ρ_0$, 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 $ρ$ is to $ρ_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neighborhood of $ρ_0$ 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 $ρ$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $ρ_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE Brutsche, Johannes Rohde, Angelika Statistics Theory Probability For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $ρ_0$, 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 $ρ_0$, 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 $ρ$ is to $ρ_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neighborhood of $ρ_0$ 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 $ρ$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $ρ_0$. |
| title | The level of self-organized criticality in oscillating Brownian motion: $n$-consistency and stable Poisson-type convergence of the MLE |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2503.07022 |