Is there any known (asymptotic) statistical test for the null hypothesis $$\|p-q\|\le \epsilon$$ for $k$-dimensional categorical data independently taken from two societies for some given norm $\|\cdot\|$ or a specifi norm such as $l_p$, $l_2$, or $l_\infty$.
When $\epsilon=0$, for any norm the null hypothesis beomes $p_i=q_i, i=1,...,k$, which has been studied in the litrature, for example, one can use the chi-sqaured test. For $k=2$, there are also two sample binomial tests.