## Bernoulli

• Bernoulli
• Volume 20, Number 2 (2014), 846-877.

### Continuous mapping approach to the asymptotics of $U$- and $V$-statistics

#### Abstract

We derive a new representation for $U$- and $V$-statistics. Using this representation, the asymptotic distribution of $U$- and $V$-statistics can be derived by a direct application of the Continuous Mapping theorem. That novel approach not only encompasses most of the results on the asymptotic distribution known in literature, but also allows for the first time a unifying treatment of non-degenerate and degenerate $U$- and $V$-statistics. Moreover, it yields a new and powerful tool to derive the asymptotic distribution of very general $U$- and $V$-statistics based on long-memory sequences. This will be exemplified by several astonishing examples. In particular, we shall present examples where weak convergence of $U$- or $V$-statistics occurs at the rate $a_{n}^{3}$ and $a_{n}^{4}$, respectively, when $a_{n}$ is the rate of weak convergence of the empirical process. We also introduce the notion of asymptotic (non-) degeneracy which often appears in the presence of long-memory sequences.

#### Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 846-877.

Dates
First available in Project Euclid: 28 February 2014

https://projecteuclid.org/euclid.bj/1393594008

Digital Object Identifier
doi:10.3150/13-BEJ508

Mathematical Reviews number (MathSciNet)
MR3178520

Zentralblatt MATH identifier
1303.60019

#### Citation

Beutner, Eric; Zähle, Henryk. Continuous mapping approach to the asymptotics of $U$- and $V$-statistics. Bernoulli 20 (2014), no. 2, 846--877. doi:10.3150/13-BEJ508. https://projecteuclid.org/euclid.bj/1393594008

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#### Supplemental materials

• Supplementary material: Supplement to paper “Continuous mapping approach to the asymptotics of U- and V-statistics”. The supplement Beutner and Zähle [7] contains a discussion of some extensions and limitations of the approach presented in this paper.