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We consider normalizing sequences that can give rise to nondegeneratelimittheorems for Birkhoff sums under the iteration of a conservative map.Mostclassical limit theorems involve normalizing sequences that arepolynomial,possibly with an additional slowly varying factor. We show that,ingeneral, there can be no nondegenerate limit theorem with anormalizingsequence that grows exponentially, but that there are examples whereitgrows like a stretched exponential, with an exponent arbitrarily close to 1.

Growth of normalizing sequences in limit theorems for conservative maps