  There is a wide literature on change point tests, but the case of variables
with infinite variances is essentially unexplored. In this paper we address
this problem by studying the asymptotic behavior of trimmed CUSUM statistics.
We show that in a location model with i.i.d. errors in the domain of attraction
of a stable law of parameter $0<\alpha <2$, the appropriately trimmed CUSUM
process converges weakly to a Brownian bridge. Thus, after moderate trimming,
the classical method for detecting change points remains valid also for
populations with infinite variance. We note that according to the classical
theory, the partial sums of trimmed variables are generally not asymptotically
normal and using random centering in the test statistics is crucial in the
infinite variance case. We also show that the partial sums of truncated and
trimmed random variables have different asymptotic behavior. Finally, we
discuss resampling procedures which enable one to determine critical values in
the case of small and moderate sample sizes.
