  We examine the performance of six estimators of the power-law
cross-correlations -- the detrended cross-correlation analysis, the detrending
moving-average cross-correlation analysis, the height cross-correlation
analysis, the averaged periodogram estimator, the cross-periodogram estimator
and the local cross-Whittle estimator -- under heavy-tailed distributions. The
selection of estimators allows to separate these into the time and frequency
domain estimators. By varying the characteristic exponent of the
$\alpha$-stable distributions which controls the tails behavior, we report
several interesting findings. First, the frequency domain estimators are
practically unaffected by heavy tails bias-wise. Second, the time domain
estimators are upward biased for heavy tails but they have lower estimator
variance than the other group for short series. Third, specific estimators are
more appropriate depending on distributional properties and length of the
analyzed series. In addition, we provide a discussion of implications of these
results for empirical applications as well as theoretical explanations.
