Let X ∼ tα be a standard (mean 0, scale 1) Student-t with ν=α degrees of freedom and let Y=|X|. Then Y has the half‑t (a.k.a. folded t) density you wrote: g(y) = 2 Γ((α+1)/2) / (√(απ) Γ(α/2)) (1 + y^2/α)^{-(α+1)/2}, y ≥ 0.

The sum Sn = ∑i=1..n Yi is the n-fold convolution of this half‑t density.

1. Closed form? In general, no

• The family of half‑t distributions is not closed under convolution. For n ≥ 2 there is no simple closed-form pdf/cdf for Sn in terms of elementary functions (this is true even for the half‑Cauchy case α=1). One can write expressions in terms of very general special functions (e.g., Fox H/Meijer G), but these are not tractable in practice.

2. Characteristic-function approach (numerical inversion)

• If CY(t) denotes the characteristic function of Y, then the CF of Sn is CSn(t) = [CY(t)]^n.

• You can recover the distribution numerically by Fourier inversion. Two standard formulas:

  • pdf: fSn(s) = (1/2π) ∫_{-∞}^{∞} e^{-its} [CY(t)]^n dt
  • cdf (Gil-Pelaez): FSn(s) = 1/2 − (1/π) ∫_0^{∞} Im{ e^{-its} [CY(t)]^n / t } dt These integrals are efficiently evaluated by FFT/quadpack-type methods with a damping/window if needed.

• For reference, the CF of X itself has a closed form: φX(t) = 2^{1−α/2} (√α |t|)^{α/2} K_{α/2}(√α |t|) / Γ(α/2), where Kν is the modified Bessel function of the second kind. The CF of Y is CY(t) = ∫_0^∞ e^{it y} g(y) dy; it does not simplify to a neat closed form for general α, but it can be evaluated numerically (e.g., via quadrature) and then used in the inversion formulas above. A numerically stable way to compute CY(t) is to use the normal–inverse-gamma scale-mixture representation of the t: conditional on the mixing variable, Y is half-normal and its CF is known in closed form, after which you integrate over the mixing distribution.

3. Useful approximations

• Moments of Y:
  • E[Y] exists iff α > 1 and equals μY = 2 √α Γ((α+1)/2) / ((α−1) √π Γ(α/2)).
  • Var(Y) exists iff α > 2 and equals σY^2 = α/(α−2) − μY^2.
• If α > 2, a normal approximation is often accurate for moderate n: Sn ≈ Normal(mean = n μY, variance = n σY^2).
• If 1 < α ≤ 2, E[Y] is finite but Var(Y) is infinite. Then (Sn − n μY)/n^{1/α} converges in distribution to an α‑stable law (totally skewed to the right).
• If 0 < α ≤ 1, even E[Y] is infinite and Sn/n^{1/α} converges to a totally skewed α‑stable law. In all heavy‑tailed cases (α ≤ 2), for tail probabilities P(Sn > s) the subexponential/regularly varying tail gives the useful approximation P(Sn > s) ≈ n P(Y > s) for large s.

4. Practical recommendation

• There is no simple closed-form distribution for Σ|Xi|. To obtain densities/cdfs, use numerical CF inversion (FFT or Gil–Pelaez) with CY(t) computed numerically, or directly perform n-fold convolution numerically (FFT on a grid). For large n, use the normal approximation when α > 2, or stable-law approximations when α ≤ 2.

If you’d like, I can outline a concrete numerical scheme (choice of grid, damping/window, step sizes) for inverting [CY(t)]^n to get fSn(s) to a desired accuracy.