  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.
