Selection from a stable box

Details Selection from a stable box Selection from a stable box

2008-03-06

arxiv.org/abs/0803.0868v1

Bernoulli 2008, Vol. 14, No. 1, 125-139

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/07-BEJ6014 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statist

Scientific paper

10.3150/07-BEJ6014

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.

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