The distribution of the sum ( S = \sum_{i=1}^n |X_i| ), where each ( X_i ) follows a symmetric ( t )-distribution with tail parameter ( \alpha ) (degrees of freedom ( \alpha )), does not generally have a closed-form expression in terms of elementary functions. Here's a detailed breakdown:

1. Folded ( t )-Distribution (PDF of ( Y = |X| ))

Given ( X \sim t_\alpha ) (symmetric ...

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. ...

Sum of Absolute Values of T Random Variables

Analysis of the Problem

You're looking at the distribution of the sum , where each follows a symmetric T distribution with 0 mean and tail parameter .

You've correctly identified that has the PDF:

First, note that because the t-distribution is symmetric around 0, the distribution of |X| (which is the folded t-distribu...

Let

  X₁ ,…, Xₙ iid ∼ tα ( centred Student-t with α>0 degrees of freedom)

and put Yᵢ = |Xᵢ|, Sₙ = Σᵢ Yᵢ .

The density of the “folded” variable

  Y = |X|

is, as you already wrote

  fY (y) = cα (α+y²)^{-ν},  y≥0, (1)

with

  ν = (α+1)/2,
  cα = 2 α^{ν}/ ( √α B(α/2 , ½) ). (2)

(Th...

If you carry the convolution through in closed‐form you will find that nothing “nice” (Gaussian, Gamma, etc.) pops out – the sum of independent half–Student–t’s can only be written either as a one–dimensional inverse Fourier (or Laplace) integral or in terms of Meijer‐G/Fox‐H functions. Below are three equivalent representations of the density of
Sₙ = ∑_{i=1}ⁿ |Xᵢ|,
where Xᵢ∼t₍α₎(0,1...

This is an interesting problem involving the sum of absolute values of t-distributed random variables. Let me work through this systematically.

Given Information

• follows a symmetric t-distribution with 0 mean and tail parameter (degrees of freedom)
• has PDF: $f_Y(y) = \frac{2 \left(\frac{\alpha }{\alpha +y^2}\right)^{\frac{\alpha +1}{2}}}{\sqrt{\alpha } B\left(\fra...

This is an excellent question. The direct approach you took using characteristic functions (CFs) is a standard technique for finding the distribution of sums of i.i.d. random variables. The reason you got stuck is that this particular problem does not have a simple, closed-form solution for the distribution of the sum for a general . The resulting characteristic function, , does not cor...

Let

 X₁ ,…, X_n iid ∼ t_ν   (ν>0) (1)

and put Y_i = |X_i | ,  S_n = Σ_{i=1}^{n} Y_i .

Except for ν=1 (the Cauchy case) and one or two other very small values of ν, the distribution of S_n does not belong to any named family and no closed-form expression involving elementary or standard special functions is known.
Below is an outline of what can be said and why a simple closed fo...