# Local asymptotic normality for shape and periodicity of a signal in the drift of a degenerate diffusion with internal variables

## TL;DR

This paper establishes Local Asymptotic Normality for a high-dimensional stochastic system with unknown shape and periodicity parameters, extending previous results to more complex models driven by low-dimensional noise.

## Contribution

It proves LAN jointly in shape and periodicity parameters for a class of degenerate diffusions with internal variables, generalizing known results to higher dimensions and more complex signals.

## Key findings

- LAN holds jointly in shape and periodicity parameters
- Local scales are n^{-1/2} for shape and n^{-3/2} for periodicity
- Results extend known LAN properties to complex, high-dimensional systems

## Abstract

Taking a multidimensional time-homogeneous dynamical system and adding a randomly perturbed time-dependent deterministic signal to some of its components gives rise to a high-dimensional system of stochastic differential equations which is driven by possibly very low-dimensional noise. Equations of this type are commonly used in biology for modeling neurons or in statistical mechanics for certain Hamiltonian systems. Assuming that the signal depends on an unknown shape parameter $\theta$ and also has an unknown periodicity $T$, we prove Local Asymptotic Normality (LAN) jointly in $\theta$ and $T$ for the statistical experiment arising from (partial) observation of this diffusion in continuous time. The local scale turns out to be $n^{-1/2}$ for $\theta$ and $n^{-3/2}$ for $T$ which generalizes known results for simpler systems.