I would like to recover the probability mass function (pmf) from the characteristic function (CF) of a discrete probability distribution using Mathematica.
Ideally, I'd like to do calculations like this example (not necessarily this simple). To compute the pmf fX+X[x] of the sum of two iid discrete uniform distributions X with support S={1,...,6}, it seems reasonable to try something like this:
Subscript[\[CurlyPhi], X + X][t_] := CharacteristicFunction[DiscreteUniformDistribution[{1, 6}], t]^2 Subscript[f, X + X][x_] := InverseFourierTransform[Subscript[\[CurlyPhi], X + X][t], t, x]
but InverseFourierTransform doesn't recover the pmf from the CF of discrete distributions (it does recover pdfs from the CFs of continuous distributions). Am I using the wrong function? (There are other candidates, but InverseFourier only works on lists of numbers, and InverseZTransform doesn't seem to work here either.) Am I forgetting to set some necessary options to the inverse function? Or is there just no built-in to recover the pmf from the CF of a discrete probability distribution?