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We prove that near epoch dependence (McLeish, 1975; Gallant and White, 1988) and<jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>-approximability (P\u00f6tscher and Prucha, 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy-tailed GARCH processes and distributed lags. 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