A general approach to the joint asymptotic analysis of statistics from sub-samples

TL;DR

This paper introduces a novel, unified method for analyzing the joint asymptotic behavior of statistics derived from sub-samples in time series, simplifying complex proofs and extending to various statistics including self-normalized and sub-sampling p-values.

Contribution

It provides a general technique combining probabilistic and analytic steps to establish joint asymptotic distributions, applicable to a wide range of sub-sample based statistics.

Findings

Unified treatment of asymptotic distributions for various statistics

Extension to self-normalized statistics and sub-sampling p-values

Results on bootstrap consistency and compact differentiability

Abstract

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.