This paper may help (I cannot easily access the full paper, so I am not certain.)

Stefanescu, Stefan V. "Generating uniform random points inside a cone." Monte Carlo Methods and Applications 6.2 (2000): 115-130. (Journal link.)

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[![ConeAbs][1]][1]

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The paper reviews previous work, so its bibliography may be helpful as well.

This paper may help (I cannot easily access the full paper, so I am not certain, but certainly the title and abstract indicate it is relevant.)

Stefanescu, Stefan V. "Generating uniform random points inside a cone." Monte Carlo Methods and Applications 6.2 (2000): 115-130. (Journal link.)

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[![ConeAbs][1]][1]

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The paper reviews previous work, so its bibliography may be useful as well.