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I need to compute probabilities of the form $P( \Vert X \Vert_\infty < r ),$ where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere $\mathcal{S}_{n-1}$. Clearly, the distribution of $Z = \Vert X \Vert_\infty$ is supported by the interval $[\frac{1}{\sqrt{n}},1]$.

Any hint to get a simple formula (ideally a closed-form expression or a one-dimensional integral) will be greatly appreciated !

Note: Using the fact that $X \sim \frac{Y}{\Vert Y\Vert_2}$ for $Y \sim \mathcal{N}(0,1)$, at first I thought I could first compute the probabilities $P_i:=P( |X_i| < r \Vert X \Vert_2)$, which can be expressed as a quantile of the F-distribution, but then I realized that those events are not independent, so $P( \Vert X \Vert_\infty < r ) \neq \prod_{i=1}^n P_i$.

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    $\begingroup$ If $r>\frac{1}{\sqrt{2}}$ we have $$P(\|X\|_\infty>r)=nP(X_1>r)=n\int_r^1 \sqrt{1-t^2}^{n-2}C(n)dt.$$ For $r<\frac{1}{\sqrt{2}}$ one could use inclusion-exclusion principle to compute probabilities. However I doubt there exist a closed form solution. $\endgroup$ – user35593 Jul 6 '15 at 12:12
  • $\begingroup$ Thanks ! This a a nice trick for the case $r>\frac{1}{\sqrt{2}}$ indeed. What is the function $C(n)$ in the integral ? $\endgroup$ – guigux Jul 7 '15 at 8:44
  • $\begingroup$ $C(n)$ is the measure of the $n-2$-dimensionale unit sphere. $\endgroup$ – user35593 Jul 7 '15 at 19:41
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There is a trick to reduce these kinds of questions to questions about independent normals, as in my ancient preprint. For large $n,$ concentration of measure will presumably give you easy estimates.

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    $\begingroup$ I'm not sure how to use your result, because the function $\mathbf{1}_{\{\Vert x \Vert_\infty < r\}} (x)$ is not homogeneous of degree $d$ for any $d$ ? $\endgroup$ – guigux Jul 7 '15 at 8:46

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