  In time series analysis, statistics based on collections of estimators
computed from sub-samples play a crucial role in an increasing variety of
important applications. Proving results about the joint asymptotic distribution
of such statistics is challenging since it typically involves a nontrivial
verification of technical conditions and tedious case-by-case asymptotic
analysis. In this paper, we provide a novel technique that allows to circumvent
those problems in a general setting. Our approach consists of two major steps:
a probabilistic part which is mainly concerned with weak convergence of
sequential empirical processes, and an analytic part providing general ways to
extend this weak convergence to functionals of the sequential empirical
process. Our theory provides a unified treatment of asymptotic distributions
for a large class of statistics, including recently proposed self-normalized
statistics and sub-sampling based p-values. In addition, we comment on the
consistency of bootstrap procedures and obtain general results on compact
differentiability of certain mappings that seem to be of independent interest.
