• Bernoulli
  • Volume 14, Number 1 (2008), 125-139.

Selection from a stable box

Alexander Aue, István Berkes, and Lajos Horváth

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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 $α$-stable law, $α∈(0, 2)$. While the functional CUSUM statistics itself converges to an $α$-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.

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Bernoulli, Volume 14, Number 1 (2008), 125-139.

First available in Project Euclid: 8 February 2008

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CUSUM functional limit theorems order statistics permutation principle stable distributions


Aue, Alexander; Berkes, István; Horváth, Lajos. Selection from a stable box. Bernoulli 14 (2008), no. 1, 125--139. doi:10.3150/07-BEJ6014.

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