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In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of Pr(Z_n>t_n n) for all sequences (t_n)_{n\\in\\N} satisfying a certain lower bound. This complements previous results, where it was assumed that Y_z has exponential moments of all orders. 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In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of Pr(Z_n>t_n n) for all sequences (t_n)_{n\\in\\N} satisfying a certain lower bound. This complements previous results, where it was assumed that Y_z has exponential moments of all orders. 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