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Cyclic and alternating $U$-statistics

Svante Janson

TL;DR

This work develops a comprehensive theory for cyclic and alternating $U$-statistics of order $2$ under i.i.d. inputs, revealing a rich dichotomy between nondegenerate normal limits and degenerate non-normal limits that depend on the decomposition into symmetric and antisymmetric parts. Using Hoeffding's decomposition and spectral analysis of associated integral operators, the authors derive explicit limit representations as sums of independent building blocks ($\zeta^2-1$, $\eta$, $\vartheta$) with eigenvalues of $T_{\widehat f-\mu}$ driving the limits. The cyclic variants exhibit decoupling of symmetric and antisymmetric contributions, while the standard and many alternating variants do not, leading to diverse limit structures including Wiener–Itô type chaos and stochastic-area components. Concrete examples, notably writhe and alternating inversion numbers, anchor the theory and connect to known permutation statistics; the paper also outlines open problems for joint convergence, strong laws, and functional limit theorems. Overall, the work extends classical $U$-statistics results to cyclic and alternating frameworks, enriching probabilistic and statistical understanding of symmetry-driven limit phenomena.

Abstract

We define cyclic $U$-statistics as a variant of $U$-statistics based on variables $X_1,\dots,X_n$ that are assumed to be cyclically ordered. We also define alternating $U$-statistics where in the definition terms are summed with alternating sings (in three different ways). Only $U$-statistics of order 2 are considered. The definitions are inspired by special cases studied by Chebikin (2008) and Even-Zohar (2017) for random permutations. We show limit theorems similar to well-known results for standard $U$-statistics, but with some differences between the different versions. In particular, we find both ``nondegenerate'' normal limits and ``degenerate'' non-normal limits.

Cyclic and alternating $U$-statistics

TL;DR

This work develops a comprehensive theory for cyclic and alternating -statistics of order under i.i.d. inputs, revealing a rich dichotomy between nondegenerate normal limits and degenerate non-normal limits that depend on the decomposition into symmetric and antisymmetric parts. Using Hoeffding's decomposition and spectral analysis of associated integral operators, the authors derive explicit limit representations as sums of independent building blocks (, , ) with eigenvalues of driving the limits. The cyclic variants exhibit decoupling of symmetric and antisymmetric contributions, while the standard and many alternating variants do not, leading to diverse limit structures including Wiener–Itô type chaos and stochastic-area components. Concrete examples, notably writhe and alternating inversion numbers, anchor the theory and connect to known permutation statistics; the paper also outlines open problems for joint convergence, strong laws, and functional limit theorems. Overall, the work extends classical -statistics results to cyclic and alternating frameworks, enriching probabilistic and statistical understanding of symmetry-driven limit phenomena.

Abstract

We define cyclic -statistics as a variant of -statistics based on variables that are assumed to be cyclically ordered. We also define alternating -statistics where in the definition terms are summed with alternating sings (in three different ways). Only -statistics of order 2 are considered. The definitions are inspired by special cases studied by Chebikin (2008) and Even-Zohar (2017) for random permutations. We show limit theorems similar to well-known results for standard -statistics, but with some differences between the different versions. In particular, we find both ``nondegenerate'' normal limits and ``degenerate'' non-normal limits.
Paper Structure (21 sections, 17 theorems, 206 equations)

This paper contains 21 sections, 17 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Cyclic $U$-statistics
  3. Alternating $U$-statistics
  4. Preliminaries
  5. Some notation
  6. Hoeffding's decomposition
  7. Three distributions
  8. Background: classical $U$-statistics
  9. Cyclic $U$-statistics
  10. Bi-alternating $U$-statistics
  11. Singly alternating $U$-statistics
  12. A short summary
  13. Examples
  14. Further results and open problems
  15. Joint convergence
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $(i,j)$ and $(k,l)$ be two pairs of indices with $i\neq j$, $k\neq l$, and $\{i,j\}\neq \{k,l\}$ (i.e., $(i,j)\neq(k,l)$ and $(i,j)\neq(l,k)$). Then $f_{12}(X_i,X_j)$ and $f_{12}(X_k,X_l)$ are uncorrelated and thus $\operatorname{\mathbb E}{}\bigl[f_{12}(X_i,X_j)f_{12}(X_k,X_l)\bigr]=0$.

Theorems & Definitions (51)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 41 more