I have a question, which is necessary at one step of my research. Suppose that $X$ is a uniform random vector on the unit sphere $$S^{d-1} := \{x \in \mathbb{R}^d: \|x\|_2 = 1\}~.$$ Is there any simple anti-concentration result, which lower bounds the quantity: $$\mathbb{P}(a^\top X > b)$$ for some $a \in \mathbb{R}^d$ and $b \in \mathbb{R}$? Of course, we need to assume that $b \le \|a\|_2$ for the event $a^\top X > b$ to be non-empty. I know that there are multiple concentration inequalities for projections of unit-norm vectors on linear subspaces, but this is not really a projection, and moreover, I want an anti-concentration result. Any help will be greatly appreciated!
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$\begingroup$ WLOG assume $\|a\|=1$ and $b=b_0/\sqrt d$. Write $X=g/\|g\|$ with $g\sim N(0,I_d)$. Then your event has the form $\{Z>b_0 \sqrt{\chi^2_d/d} \}$ with $Z=g^Ta \sim N(0,1)$ and $\chi^2_d$ chi-square with $d$ degrees of freedom. Exponential concentration of the chi-square should give a solution. $\endgroup$– jlewkCommented Aug 1, 2021 at 10:23
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$\begingroup$ @jlewk : Note that $Z$ and $\chi_d^2$ are dependent. $\endgroup$– Yuval PeresCommented Aug 1, 2021 at 17:28
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$\begingroup$ Yes; to overcome dependence I had in mind to work instead with $b_0(1\pm\epsilon)$ in the event $|\sqrt{\chi^2_d/d}-1|\le \epsilon$ that would not affect the probability of the event by more than $2e^{-\epsilon^2 d/2}$. $\endgroup$– jlewkCommented Aug 1, 2021 at 18:34
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1 Answer
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Replacing $b$ by $b/\|a\|_2$, we may assume WLOG that $\|a\|_2=1$. Let $R$ be a rotation matrix that maps $a$ to $v=(1,0,\ldots,0)^T$. Then $a^Tx=R(a)^T R(X)=v^T R(X)$, which has the same distribution as $v^T X=X_1$. Thus (assuming $\|a\|_2=1$) we have that $\mathbb P(a^TX>b)=\mathbb P(X_1>b)$ is the area of the cap $\{x \in S^{d-1} : x_1>b\}$ on the unit sphere, divided by the total are of the sphere. There are explicit integral formulas for the area of caps, see e.g. [1] and the first answer in [2], but perhaps most useful is the discussion in Chapter 2 of [3], as noted in the second answer in [2].
[1] S. Li, Concise formula for the area and volume of a hyperspherical cap, Asian J. of Math. and Stat. (2011), http://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf
[2] https://math.stackexchange.com/questions/1251916/surface-area-of-a-section-of-the-unit-sphere
[3] Keith Ball, An Elementary Introduction to Modern Convex Geometry http://library.msri.org/books/Book31/files/ball.pdf
answered Aug 1, 2021 at 18:28