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

中文翻译：

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独立同分布随机变量平均值矩的对数凹和对数凸

我们表明，iid 正随机变量的平均值小于 1 的矩序列是对数凹的。对于至少 1 阶的矩，我们推测该序列是对数凸的，并表明这最终适用于整数矩（在忽略序列的前 $p^2$ 项之后）。