Degenerate $$U$$- and $$V$$-statistics under ergodicity: asymptotics, bootstrap and applications in statistics

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Energy statistics: A class of statistics based on distances (2013, background): “For general Gaussian distance covariance (Definition 7), it makes sense to choose the Gaussian processes with ϕðtÞ ¼ VarðX1þ⋯þXtÞ, which implies that for the corresponding covariance function kðs,tÞ we have 2kðs,tÞ ¼ 2E½XðtÞXðsÞ ¼ E½XðtÞ2 þE½XðsÞ2 −E½XðtÞ−XðsÞ 2 ¼ ϕðtÞþϕðsÞ−ϕðt−sÞ: We also have the ”