non stationarity is sometimes considered in the sense of econometricians as unit root problems 5PCB Philipps et al.).
A question would be to unify "reasonable" notions of non stationarity, eg local stationarity (eg Dahlhaus and followers), periodic or seasonal (eg Dickey, Mohamedou Ould Haye, Viano, Leskow), unit roots.
For periodic behaviors an additional attractive question is to fit periods (even if the continuous time case is even more relevant and a problem is then to fit periods
Many other non stationarity notions, such as random walks in random environment (eg Snitzmann) are extremely attractive: Anyway they dont necessarily features of real data
The idea of the present project is more to develop (non parametric or parametric) models featuring the main properties of real data sets to be fitted
Certainly the validity of techniques should be provided from real data analysis as this was emblematically done in a contradictory paper by Mikosch and Starica opposing long range dependent models to models with a linear trend
Nonstationarity in a broad sense means that the marginals and/or finite dimensional distributions of the process {Xt, t ∈ T } depend on the index "t". de-trending, de-seasonalization, ARIMA modeling ARCH, GARCH modelling etc are ways of handling the dependence of time on the series.
The question set is not only at the mathematical level, I think that my publications express that I am more or less aware of the technique
The problem is to fit data, thus I want to extract the important features of non stationary models in order to propose adequate families of models
Applied viewpoints seem more relevant in this case
Thanks for being more ellaborate. As a teacher I usually assign real data analysis to students. As and when I come accross some atypical behaviour, I will communicate with you.
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