You could probably pull out the exact probability mass function with characteristic functions. To review why these would be useful, let χ[T](s) = ⟨ exp(i ω T)

You could probably pull out the exact probability mass function with characteristic functions. To review why these would be useful, let χ[T](s) = ⟨ exp(i ω T) ⟩, then χ[A + B] = χ[A] χ[B] for any A and B which are independent, so the characteristic function for a uniform random variable is going to be χ = (exp(i ω) − 1)/(i ω) and thus the sum of n independent uniform random numbers is just χⁿ. Meanwhile the PDF of a random variable is just the inverse Fourier transform of the characteristic function, so we have fₙ(t) = ∫ dω/(2π) exp(-i ω t) [χ(ω)]ⁿ Pr(n samples > 1) = ∫{1 → ∞} dt fₙ(t). You could then interchange those two integrals, solving the convergence problem of exp(-i ω ∞) by giving ω an infinitesimal negative-imaginary component. This translates to moving the ω contour a little below the real line, and then the normal tricks of complex analysis should give you definite values for all of the integrals as residues about the simple pole at 0.