Let {Xj} 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 {Xj} 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 {Xj} and the permutation are random, the situation turns out to be more delicate if a realization of the {Xj} 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.