I'd like to go on record saying that the distribution of the sum of uniform distributions is not fun to work with 
You can probably do it algebraically using multiple variable integrals of the p.d.f. (uniform distribution is P(x) = H(x) +(-1)H(x-1) ) so you end up with something like integral y = -inf to inf P(x)P(y) dy But just the fact that it's a piecewise function makes it annoying to work with.
Which, in 2 variables, just ends up being a triangle. P(x) = xH(x) - 2(x-1)H(x-1) + (x-2)H(x-2) But that limits you to finding 2hko probabilities. As a diagram, this would be like the diagonal area under a square, where the square has uniform height.
In the long run just running a few thousand simulations will probably get you close enough, and be a lot easier to write.
You can probably do it algebraically using multiple variable integrals of the p.d.f. (uniform distribution is P(x) = H(x) +(-1)H(x-1) ) so you end up with something like integral y = -inf to inf P(x)P(y) dy But just the fact that it's a piecewise function makes it annoying to work with.
Which, in 2 variables, just ends up being a triangle. P(x) = xH(x) - 2(x-1)H(x-1) + (x-2)H(x-2) But that limits you to finding 2hko probabilities. As a diagram, this would be like the diagonal area under a square, where the square has uniform height.
In the long run just running a few thousand simulations will probably get you close enough, and be a lot easier to write.