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Added line about other distributions
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Mark Meckes
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For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.

The paper also shows how to generate certain other distributions on the $\ell_p^n$ ball by modifying the distribution of $Y$.

For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.

For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.

The paper also shows how to generate certain other distributions on the $\ell_p^n$ ball by modifying the distribution of $Y$.

Source Link
Mark Meckes
  • 11.4k
  • 3
  • 38
  • 69

For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.