The α-Cauchy variable C_α is infinitely divisible if and only if 1 < α ≤ 2.
On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line.Scand
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Infinite divisibility of $\alpha$-Cauchy distributions
The α-Cauchy variable C_α is infinitely divisible if and only if 1 < α ≤ 2.