Copula used for pseudo-random generation from continuous variables only?

Question

I know multivariate pseudorandom generation from continuous distributions can be done using copulas and inverse transform sampling.

The question is if copula have application in "discrete" pseudo-random generation also?

Answer 1

In principle the inverse transform method can be used for discrete distributions as well. This boils down to first subdividing the interval $[0,1]$, where the sub-intervals are formed according to the distribution function $F_X$ of the target random variable $X$. Then a standard uniform random variable $U$ is drawn and one assigns a value to $X$ depending on the sub-interval in which $U$ falls. A more technical explanation can for example be found in http://www.columbia.edu/~ks20/4404-Sigman/4404-Notes-ITM.pdf (or most introductory probability / statistics text books). As noted in the pdf this is not always the most convenient way to go about (for example in the Poisson case).

Once a sampling algorithm based on a uniform random variable is available a multivariate observation can be generated via copulas as usual. First a pseudo-realization $\mathbf{U} = (U_1,\ldots,U_d)$ is drawn from the copula. Then you generate $X_i$ from $U_i$ (as described above) to get $\mathbf{X} = (X_1,\ldots,X_d)$.