Multiple discrete RV convolutions performance

I have a discrete random variable with PDF of (just a simplified example)

Piecewise[{{1/3, x == 5 || x == 10 || x == 15}}, 0] 

Actually, in my code this is a MixtureDistribution of a number of discrete distributions.

What I need is to obtain a multiple convolution (for example a 100-fold) of independent identically distributed random variables, distributed according to the provided distribution.

I tried all known to me ways:

1) TransformedDistribution

TransformedDistribution[Plus @@ Flatten[Table[tX[v], {v, 0, vMax}]], Flatten[Table[tX[v] \[Distributed] resF, {v, 0, vMax}]]] 

2) multiple DiscreteConvolve

DiscreteConvolve[PDF[resF, x], PDF[resF, x], x, y] 

where resF is the provided initial distribution.

The first method provides the needed result, but it takes way too long and the time grows exponentially - approx 60 seconds to obtain a ~10-fold convolution, a few minutes for the next.

The second method seem to provide the correct answer for a couple of first convolutions (and it take longer compared to the first method), but the next convolutions give 0.

Just to compare I wrote a simple head-on program on C# to do the task and it takes only ~2-3 second to do the 300-fold convolution. And there is way to optimize it by removing values, higher than the threshold on the go.

Ultimately, I need the probability that the final convolution will be smaller than a provided threshold, i.e. CDF.

Can someone tell me why is it taking so long or if I'm doing this the wrong way.

Thanks!

EDIT

Some code that I test:

ClearAll[resFF, vMax]; AbsoluteTiming[ resF = EmpiricalDistribution[{1/3, 1/3, 1/3} -> {5, 10, 15}]; vMax = 10; PDF[TransformedDistribution[ Plus @@ Flatten[Table[tX[v], {v, 1, vMax}]], Flatten[Table[tX[v] \[Distributed] resF, {v, 1, vMax}]]], x] ] 

Here resF is the distribution I mentioned in the beginning, and vMax is the number of convolutions I want to obtain. So this code gives the PDF of a 10-fold convolution.

On my machine it takes 3 seconds to do the 10-fold convolution, 8 seconds for 11-fold and 25 seconds for 12-fold. As I need more than 100-fold one, this performance is not sufficient.

EDIT 2

As Jim Baldwin said, I missed the fact that I can obtain the result using the generating functions of discrete random variables.

Thus, the code I came up with that gives me the desired value of vMax-fold convolution CDF in point 1000 is:

ClearAll[resF, gen, vMax]; vMax = 150; resF[x_] := Piecewise[{{1/3, x == 5 || x == 10 || x == 15}}, 0]; gen = GeneratingFunction[resF[n], n, x]; Total@Select[CoefficientList[(gen)^vMax, x][[1 ;; 1000]], # > 0 &] 

However, in this case I have to scale the initial data, if the outcomes of RV are not integer.

PS: This is out of my own curiosity, but still - if anyone know why the TransformedDistribution solution works so slow, please, tell me. Thanks!