  Data objects taking value in a general metric space have become increasingly
common in modern data analysis. In this paper, we study two important
statistical inference problems, namely, two-sample testing and change-point
detection, for such non-Euclidean data under temporal dependence. Typical
examples of non-Euclidean valued time series include yearly mortality
distributions, time-varying networks, and covariance matrix time series. To
accommodate unknown temporal dependence, we advance the self-normalization (SN)
technique (Shao, 2010) to the inference of non-Euclidean time series, which is
substantially different from the existing SN-based inference for functional
time series that reside in Hilbert space (Zhang et al., 2011). Theoretically,
we propose new regularity conditions that could be easier to check than those
in the recent literature, and derive the limiting distributions of the proposed
test statistics under both null and local alternatives. For change-point
detection problem, we also derive the consistency for the change-point location
estimator, and combine our proposed change-point test with wild binary
segmentation to perform multiple change-point estimation. Numerical simulations
demonstrate the effectiveness and robustness of our proposed tests compared
with existing methods in the literature. Finally, we apply our tests to
two-sample inference in mortality data and change-point detection in
cryptocurrency data.
