Are fractional/continuous central moments useful? If the $k$th central moment for a continuous probability density function $f(x)$ is is defined by
$$
m_k = \int (x-\mu)^k f(x) dx
$$
We get the variance for $m_2$ and the unstandardized skew, kurtosis, etc... for $k=3,4,...$ Does it make sense to go from $k\in\mathbb{N}$ and generalize to fractions or real numbers $k\in\mathbb{Q^+}$ or $k\in\mathbb{R^+}$? 
To keep this an objective question, I'm looking for an application or use of fractional or continuous moments, either as defined above or in the spirit of the question.
 A: If we define the $p$-norm of a random variable $\mathbf X$ as $\|\mathbf X\|_p = \mathbb E[|\mathbf X|^p]^{1/p}$ (this is a function of basically the raw $p^{\text{th}}$ moment of $\mathbf X$, ignoring the absolute value), then Hölder's inequality tells us that $$\mathbb E[\mathbf X \mathbf Y] \le \|\mathbf X\|_p \|\mathbf Y\|_q$$ for any $p, q \in [1,\infty)$ with $\frac1p + \frac1q = 1$. One purpose of this inequality in probability theory is to "uncouple" two random variables $\mathbf X$ and $\mathbf Y$. If they are not independent, but their distributions are easy to understand individually, then the right-hand side of Hölder's inequality may be used as an easier-to-work-with upper bound on the left-hand side.
Now, the natural choice of $p$ and $q$ is to take $p=q=2$, in which case we just get back the Cauchy-Schwarz inequality. This treats $\mathbf X$ and $\mathbf Y$ identically. We get more flexibility if we vary $p$ and $q$: maybe the higher moments of $\mathbf X$ are very badly behaved, and not so much for $\mathbf Y$, so we get better upper bounds if $q$ is large. But if we do vary $p$ and $q$, we're stuck taking a fractional moment of at least one of our random variables, possibly both.
A: One example: the stable law with $\alpha\in(0,2)$ has no moments of integer order greater than $1$ (for $\alpha\in(0,1]$ even the expectation does not exist). So in order to measure the "dispersion" one should look at the moments of lower order. Say, for $\alpha\in(1,2)$ this would be $E[|\xi-\mu|^p]$ with $p\in(0,\alpha)$.
