Selection from a stable box

TL;DR

This paper investigates the asymptotic behavior of CUSUM statistics for i.i.d. variables in the domain of attraction of a stable law, revealing complex convergence properties when the data is fixed and only the permutation varies.

Contribution

It extends the understanding of CUSUM statistics' convergence to stable laws, especially under fixed data and random permutation scenarios.

Findings

CUSUM statistics converge to an $ ext{α}$-stable bridge.

Permuted CUSUM converges to a random, nondegenerate limit.

Behavior differs when data is fixed versus when both data and permutation are random.

Abstract

Let $\{X_j\}$ be independent, identically distributed random variables. It is well known that the functional CUSUM statistic and its randomly permuted version both converge weakly to a Brownian bridge if second moments exist. Surprisingly, an infinite-variance counterpart does not hold true. In the present paper, we let $\{X_j\}$ be in the domain of attraction of a strictly $\alpha$-stable law, $\alpha\in(0,2)$. While the functional CUSUM statistics itself converges to an $\alpha$-stable bridge and so does the permuted version, provided both the $\{X_j\}$ and the permutation are random, the situation turns out to be more delicate if a realization of the $\{X_j\}$ is fixed and randomness is restricted to the permutation. Here, the conditional distribution function of the permuted CUSUM statistics converges in probability to a random and nondegenerate limit.