  The Amazon rainforest is considered one of the Earth's tipping elements and
may lose stability under ongoing climate change. Recently a decrease in
tropical rainforest resilience has been identified globally from remotely
sensed vegetation data. However, the underlying theory assumes a Gaussian
distribution of forest disturbances, which is different from most observed
forest stressors such as fires, deforestation, or windthrow. Those stressors
often occur in power-law-like distributions and can be approximated by
$\alpha$-stable L\'evy noise. Here, we show that classical critical slowing
down indicators to measure changes in forest resilience are robust under such
power-law disturbances. To assess the robustness of critical slowing down
indicators, we simulate pulse-like perturbations in an adapted and conceptual
model of a tropical rainforest. We find few missed early warnings and few false
alarms are achievable simultaneously if the following steps are carried out
carefully: First, the model must be known to resolve the timescales of the
perturbation. Second, perturbations need to be filtered according to their
absolute temporal autocorrelation. Third, critical slowing down has to be
assessed using the non-parametric Kendall-$\tau$ slope. These prerequisites
allow for an increase in the sensitivity of early warning signals. Hence, our
findings imply improved reliability of the interpretation of empirically
estimated rainforest resilience through critical slowing down indicators.
