Distribution of the Volume of Weighted Gaussian Simplex

Let $X_0, \ldots, X_l$ be independent standard Gaussian vectors in $\mathbb{R}^d$ such that $l \leqslant d$. We derive an explicit formula for the distribution of the volume of weighted Gaussian simplex without the origin -- $l$-dimensional simplex $\mathrm{conv}(σ_0X_0, \ldots, σ_lX_l)$ ($σ_0, \ldots, σ_l > 0$).

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