Multiscale asymptotics for dependent data

In many applications where temporal and spatial data is encountered, nonstationarity and dependence are prevalent. For example, in a technical system, nonstationarity can be due to degradation or varying load; financial time series are usually subject to market dynamics and macroeconomical trends; and meteorological measurements are subject to climate change.
This leads to two different perspectives on spatio-temporal data:

• How to perform reliable inference _despite_ nonstationarity and dependence?
• How to perform reliable inference _about_ the nonstationarity and dependence?

In this talk, I will briefly review the framework of locally-stationary time series and then illustrate the implications for sequential analysis, specifically for changepoint testing. Optimal proceduresfor this problem make use of multiscale test statistics, but their adaptations to nonstationary and dependent errors are either infeasible or asymptotically sub-optimal. I demonstrate how to resolve this problem via the novel asymptotic concept of thresholded weak convergence. The new approach also streamlines multiscale methodology for related problems, e.g. to segment sequences of regression curves or to provide confidence statements for relevant changepoints. 

In the second part of the talk, I discuss how nonparametric rates of convergence can arise in parametric problems if several parameters are separated in scale. Statistical lower bounds for arrays of time series are presented which allow for a unified treatment of these phenomena in Gaussian processes.