Two random variables are defined as subindependent if their covariance is zero--in other words, if they are uncorrelated. The latter link notes that "not all uncorrelated variables are independent. For example, if $X$ is a continuous random variable uniformly distributed on $[−1, 1]$ and $Y = X^2$, then $X$ and $Y$ are uncorrelated even though $X$ determines $Y$ and a particular value of $Y$ can be produced by only one or two values of $X$." So subindependence, as you can guess from the name, is a weak form of independence.
Soon after reading this, I was looking at Pearson's chi-square test for independence. The wikipedia page says that "for the test of independence, a chi-square probability of less than or equal to 0.05 (or the chi-square statistic being at or larger than the 0.05 critical point) is commonly interpreted by applied workers as justification for rejecting the null hypothesis that the row variable is independent of the column variable. The alternative hypothesis corresponds to the variables having an association or relationship where the structure of this relationship is not specified." A previous CV answer (here) also indicates that this is what the chi-square test is looking for.
Now, how can you test a null hypothesis of independence by looking for a correlation but not define the lack of a correlation as indicative of independence? Granted, my skepticism is based on my reading of the Wikipedia pages for these concepts, which could easily be flawed. But it seems to me like Pearson's chi-square method must be testing for the lack of subindependence. Is there something wrong with my conclusions? Or, is this already common knowledge?