I have a question concerning the convolution of generalized hyperbolic distributions. Proposition 6.13 of McNeil, Embrechts, Frey states the following:

If $X$ has a $d$-dimensional generalized hyperbolic distribution $$ X \sim GH_d(\lambda, \chi, \psi, \mu, \Sigma, \gamma), $$ where $\lambda, \chi, \psi \in \Bbb R$, $\mu, \gamma \in \Bbb R^d$ and $\Sigma \in \Bbb R^{d \times d}$, then for $B \in \Bbb R^{k \times d}$ we have that

$$ BX \sim GH_k(\lambda, \chi, \psi, B\mu, B\Sigma B', B\gamma) $$

As a special case of this proposition we get that the sum of two independent one- dimensional generalized hyperbolic distributions with the same parameters $\lambda, \chi, \psi$ has again a generalized hyperbolic distribution: $$ X_i \sim GH_1(\lambda, \chi, \psi, \mu_i, \sigma_i^2, \gamma_i) \implies X_1 + X_2 \sim GH_1(\lambda, \chi, \psi, \mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2, \gamma_1 + \gamma_2). $$ This follows from taking $B = (1,1)$.

But numerical experiments show that this does not seem to be the case. Also in this article the author claims that only certain sub distributions of the Generalized hyperbolic distribution are closed under convolution.

So there has to be something I am missing or getting wrong. Any help is appreciated.


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