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dohmatob
dohmatob
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Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally distributed with variance $R^2/n$, but I'm not quite sure.

Question

More formaly, if $\Phi$ is the CDF of the standard Guassian $\mathcal N(0, 1)$, what is a good upper bound for the quantity $\alpha_n := \sup_{z \in \mathbb R}|P(X_1 \le nR^{-2}z) - \Phi(z)|$ ?

Observations

My wild guess is that $\alpha_n \le Cn^{-1/2}$ for some absolute constant $C$ independent of $n$ and $R$.

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally distributed with variance $R^2/n$, but I'm not quite sure.

Question

More formaly, if $\Phi$ is the CDF of the standard Guassian $\mathcal N(0, 1)$, what is a good upper bound for the quantity $\alpha_n := \sup_{z \in \mathbb R}|P(X_1 \le nR^{-2}z) - \Phi(z)|$ ?

Observations

My wild guess is that $\alpha_n \le Cn^{-1/2}$ for some constant $C$ independent of $n$ and $R$.

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally distributed with variance $R^2/n$, but I'm not quite sure.

Question

More formaly, if $\Phi$ is the CDF of the standard Guassian $\mathcal N(0, 1)$, what is a good upper bound for the quantity $\alpha_n := \sup_{z \in \mathbb R}|P(X_1 \le nR^{-2}z) - \Phi(z)|$ ?

Observations

My wild guess is that $\alpha_n \le Cn^{-1/2}$ for some absolute constant $C$ independent of $n$ and $R$.

Source Link
dohmatob
dohmatob
  • 6.9k
  • 1
  • 18
  • 76

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally distributed with variance $R^2/n$, but I'm not quite sure.

Question

More formaly, if $\Phi$ is the CDF of the standard Guassian $\mathcal N(0, 1)$, what is a good upper bound for the quantity $\alpha_n := \sup_{z \in \mathbb R}|P(X_1 \le nR^{-2}z) - \Phi(z)|$ ?

Observations

My wild guess is that $\alpha_n \le Cn^{-1/2}$ for some constant $C$ independent of $n$ and $R$.