## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 8, Number 1 (2014), 390-431.

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

Stanislav Volgushev and Xiaofeng Shao

#### Abstract

In time series analysis, statistics based on collections of estimators computed from subsamples 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 are of independent interest.

#### Article information

**Source**

Electron. J. Statist., Volume 8, Number 1 (2014), 390-431.

**Dates**

First available in Project Euclid: 18 April 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1397826706

**Digital Object Identifier**

doi:10.1214/14-EJS888

**Mathematical Reviews number (MathSciNet)**

MR3195121

**Zentralblatt MATH identifier**

1294.62023

**Subjects**

Primary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G09: Resampling methods 62G15: Tolerance and confidence regions

Secondary: 62G20: Asymptotic properties

**Keywords**

Change point compact differentiability empirical processes self-normalization sub-sampling time series weak convergence

#### Citation

Volgushev, Stanislav; Shao, Xiaofeng. A general approach to the joint asymptotic analysis of statistics from sub-samples. Electron. J. Statist. 8 (2014), no. 1, 390--431. doi:10.1214/14-EJS888. https://projecteuclid.org/euclid.ejs/1397826706