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We show that the sequence of moments of order less than 1 of averages ofi.i.d. positive random variables is log-concave. For moments of order at least1, we conjecture that the sequence is log-convex and show that this holdseventually for integer moments (after neglecting the first $p^2$ terms of thesequence).

Log-concavity and log-convexity of moments of averages of i.i.d. random variables