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This article deals with the hypothesis test for the extremely heavy-taileddistributions with infinite mean or variance by using a truncated sample mean.We obtain three necessary and sufficient conditions under which the asymptoticdistribution of the truncated test statistics converges to normal, neithernormal nor stable or converges to $-\infty$ or the combination of stabledistributions, respectively. The numerical simulation illustrates anapplication of the theoretical results above in the hypothesis testing.

Hypothesis Test of a Truncated Sample Mean for the Extremely Heavy-Tailed Distributions