  We study properties of two resampling scenarios: Conditional Randomisation
and Conditional Permutation schemes, which are relevant for testing conditional
independence of discrete random variables $X$ and $Y$ given a random variable
$Z$. Namely, we investigate asymptotic behaviour of estimates of a vector of
probabilities in such settings, establish their asymptotic normality and
ordering between asymptotic covariance matrices. The results are used to derive
asymptotic distributions of the empirical Conditional Mutual Information in
those set-ups. Somewhat unexpectedly, the distributions coincide for the two
scenarios, despite differences in the asymptotic distributions of the estimates
of probabilities. We also prove validity of permutation p-values for the
Conditional Permutation scheme. The above results justify consideration of
conditional independence tests based on resampled p-values and on the
asymptotic chi-square distribution with an adjusted number of degrees of
freedom. We show in numerical experiments that when the ratio of the sample
size to the number of possible values of the triple exceeds 0.5, the test based
on the asymptotic distribution with the adjustment made on a limited number of
permutations is a viable alternative to the exact test for both the Conditional
Permutation and the Conditional Randomisation scenarios. Moreover, there is no
significant difference between the performance of exact tests for Conditional
Permutation and Randomisation schemes, the latter requiring knowledge of
conditional distribution of $X$ given $Z$, and the same conclusion is true for
both adaptive tests.
