If $X$ is a nonnnegative, nondegenerate random variable, the Lp norm $m(p) = E[X^p]^{1/p}$ is a strictly increasing function of $p\geq 1$. I am curious about whether it is ever possible that $\lim_{p\to\infty} \frac{m(p)}{p}$ can fail to exist. In other words, a situation where it does not equal infinity or any finite value, but rather the lim sup and lim inf fail to agree. Such a situation would imply there are oscillations in $m(p)$ but I can't think of any random variables that would produce such a moment sequence.
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