I think there is a missing concept in your question. If you are discussing simple gambles and not things like the stock market, then the distributions that are involved usually have sufficient statistics.
A statistic is sufficient for a parameter if the point estimate could be substituted for the data itself with no loss in information. A statistic, $t$, is said to be sufficient for $\theta$ if $\Pr(\mathbf{x}|\theta)=\Pr(t|\theta)$, where $\mathbf{x}$ is a vector of data. What this implies is that the implied distribution from an observed set of data has a perfect substitute in the vector of point statistics when sufficiency holds. They are indistinguishable concepts mathematically.
If you are talking about the stock market, then those distributions lack both moments and sufficient statistics and people need to use the distribution or fully process the uncertainty of the parameters using Bayesian methods as Frequentist solutions do not exist that are admissible in the general case. The reason is relatively simple.
Consider a stock traded on the NYSE. It is sold in a double auction with many potential buyers and many potential sellers. Since the stock is sold in a double auction it follows that the winner's curse does not obtain. Since the winner's curse does not obtain, it follows that the rational behavior is to bid your expectation regarding the price. Since there are many potential buyers and sellers, the limiting distribution of those prices is the normal distribution from the central limit theorem.
If we assume we are in equilibrium, then the purchase price and the selling price are both normally distributed ignoring bankruptcies, mergers and liquidity costs. Those do not change the general principle here, but do radically alter the mixture distribution. Nonetheless, if we center our graph around $(p_t^*,p_{t+1}^*)$ then we could also think of it as being $(0,0)$ in the error space. The return can be operationally defined as $$r_t=\frac{p_{t+1}}{p_t}-1,$$ so it follows that returns are the ratio of two normal distributions. In error space, the distribution of two normals centered around zero is the Cauchy distribution. If you then translate the distribution back to price space and truncate due to the limit of liability your distribution of returns must be $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$
Although you could manually verify the absence of statistics by the Neyman Fisher Factorization Theorem, it is well known that the statistics from this distribution are not sufficient. This can also be seen from the Pitman–Koopman–Darmois theorem which basically states that only distributions in the exponential family of distributions, such as the normal, have sufficient statistics. Joint sufficiency does exist for purposes of inference, but not for projective purposes which are what matter here. As such, any point estimator not constructed from the Bayesian posterior predictive density will lose information, this is amplified by the truncation.
The Bayesian posterior predictive distribution can be used because it is defined as $\Pr(\tilde{x}|\mathbf{x})$. Notice that there are no parameters at all in that probability statement.
The problem with this distribution is that neither skew nor kurtosis are even defined for it.
Consequently, higher moment discussions with stocks are deeply flawed as the above distribution has no moments at all.
Old articles such as
SCOTT, R. C. and HORVATH, P. A. (1980), On The Direction of Preference for Moments of Higher Order Than The Variance. The Journal of Finance, 35: 915-919.
depend upon the existence of moments in the distributions at all.
Unfortunately, I couldn't find specific literature to what you are talking about outside the finance literature and most of it is built on flawed assumptions. It would be interesting to see how people respond to skewed distributions like waiting time models that possess sufficiency. I couldn't find an example.