Within the context of sample-based ratio estimators (i.e., sampling without replacement), this article here stated that the covariances of sample biases tend to zero with increasing sample size $n$:
\begin{align}
\ E[(\bar x-\bar X)^r(\bar y-\bar Y)^s] & = O[\frac{1}{n^{r+s}}]\\
& = O[\frac{1}{n^{r+s+1}}]\\
\end{align}
The first big $O$ refers to when $r+s$ even whereas the second big $O$ refers to the case when $r+s$ is odd.
To be honest, the article presents proof that is rooted in polykays as well as products of multivariate symmetric means. I tried to study the proof as well as the literature related to polykays, including here and here. Even though I somewhat understand the concept behind polykays etc., I still do not get most of the technical details the above-mentioned proof is based on (I have no formal mathematical training). To me, it also seems that polykays were a topic within the research literature especially within the 1940s, 1950s, and 1960s (at least if measured in terms of some publications). I am therefore also wondering if there is any other framework besides polykays one can use to prove this statement above. Does anyone have any idea how to approach this problem without referring to polykays?