# Asymptotic nonparametric statistical analysis of stationary time series

## TL;DR

This paper reviews asymptotic nonparametric statistical methods for stationary time series, highlighting what can and cannot be achieved with stationarity assumptions alone, including clustering, change point detection, and hypothesis testing.

## Contribution

It summarizes recent results on the asymptotic consistency of algorithms for stationary time series, clarifying the limits and possibilities of statistical inference under minimal assumptions.

## Key findings

- Certain problems like homogeneity are impossible to solve under stationarity alone.
- Algorithms for clustering and change point detection can be asymptotically consistent.
- A topological criterion for the existence of consistent tests is proposed.

## Abstract

Stationarity is a very general, qualitative assumption, that can be assessed on the basis of application specifics. It is thus a rather attractive assumption to base statistical analysis on, especially for problems for which less general qualitative assumptions, such as independence or finite memory, clearly fail. However, it has long been considered too general to allow for statistical inference to be made. One of the reasons for this is that rates of convergence, even of frequencies to the mean, are not available under this assumption alone. Recently, it has been shown that, while some natural and simple problems such as homogeneity, are indeed provably impossible to solve if one only assumes that the data is stationary (or stationary ergodic), many others can be solved using rather simple and intuitive algorithms. The latter problems include clustering and change point estimation. In this volume I summarize these results. The emphasis is on asymptotic consistency, since this the strongest property one can obtain assuming stationarity alone. While for most of the problems for which a solution is found this solution is algorithmically realizable, the main objective in this area of research, the objective which is only partially attained, is to understand what is possible and what is not possible to do for stationary time series. The considered problems include homogeneity testing, clustering with respect to distribution, clustering with respect to independence, change-point estimation, identity testing, and the general question of composite hypotheses testing. For the latter problem, a topological criterion for the existence of a consistent test is presented. In addition, several open questions are discussed.