Reinhard Höpfner

Limit theorems for null recurrent Markov processes

Introduction Harris recurrence Stable increasing processes and Mittag Leffler processes The main theorem Proofs for subsection 3.1 - sufficient condition Proofs for subsection 3.1 - necessary condition Nummelin splitting in discrete time Nummelin-like splitting for general continuous time Harris processes and proofs for subsection 3.3 Overview: assumptions (H1) - (H6) References.

On a Problem of Statistical Inference in Null Recurrent Diffusions

We consider a particular example of statistical inference in null recurrent one-dimensional diffusions. In a first parametric model, we prove local asymptotic mixed normality (LAMN) and efficiency of the sequence of maximum likelihood estimates (MLE): its speed of convergence is nα/2 with α ranging over (0, 1). In a second semiparametric model (where in addition an unknown nuisance function with known compact support is included in the drift), we prove a local asymptotic minimax bound and specify asymptotically efficient estimates for the unknown parameter.

Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion

We consider a diffusion (ξt)t≥0 whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter ϑ of interest. We consider sequences of local models at ϑ corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ϑ. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ϑ, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n−3/2 at ϑ, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n−2 at ϑ, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2.

Ergodicity for a stochastic Hodgkin–Huxley model driven by Ornstein–Uhlenbeck type input

We consider a model describing a neuron and the input it receives from its dendritic tree when this input is a random perturbation of a periodic deterministic signal, driven by an Ornstein-Uhlenbeck process. The neuron itself is modeled by a variant of the classical Hodgkin-Huxley model. Using the existence of an accessible point where the weak Hoermander condition holds and the fact that the coefficients of the system are analytic, we show that the system is non-degenerate. The existence of a Lyapunov function allows to deduce the existence of (at most a finite number of) extremal invariant measures for the process. As a consequence, the complexity of the system is drastically reduced in comparison with the deterministic system.

Birth and Death on a Flow: Local Time and Estimation of a Position‐Dependent Death Rate

The paper deals with statistical inference for an unknown rate of position‐dependent killing in birth and death on a flow. We introduce local time for this process and discuss its asymptotics with the help of a Tanaka formula. As a consequence, we can prove asymptotic normality of kernel estimators for the death rate.

On frequency estimation for a periodic ergodic diffusion process

We consider the problem of frequency estimation by observations for a periodic diffusion process possessing ergodic properties in two different situations. The first corresponds to a trend coefficient continuously differentiable with respect to parameter, and the second, to a discontinuous trend coefficient. It is shown that in the first case the maximum likelihood and Bayesian estimators are asymptotically normal with rate T3/2, and in the second case these estimators have different limit distributions with rate T2.

LAMN in a class of parametric models for null recurrent diffusions

We study statistical models for one-dimensional diffusions which are null recurrent. A first parameter in the drift is the principal one, and determines regular varying rates of convergence for the score and the information process. A finite number of other parameters, of secondary importance, introduces additional flexibility for the modelization of the drift, and does not perturb the null recurrent behaviour. Under time-continuous observation we obtain local asymptotic mixed normality, state a local asymptotic minimax bound, and specify asymptotically optimal estimators.

On Minimum Distance Estimation in Recurrent Markov Step Processes II

Consider a Markov step process X = (Xt)t≥0 whose generator depends on an unknown parameters ϑ. We are interested in estimation of ϑ by a class of minimum distance estimators (MDE) based on observation of X up to time Sn, with (Sn)n a sequence of stopping times increasing to ∞. We give a precise description of the MDE error at stage n, for n fixed, i.e. a stochastic expansion in terms of powers of a norming constant and suitable coefficients (which can be calculated explicitly from the observed path of X up to time Sn).