A simplified proof of CLT for convex bodies

TL;DR

This paper offers a concise, accessible proof of Klartag's central limit theorem for convex bodies, relying solely on classical properties of log-concave functions, and includes an appendix linking thin shell conditions to CLT.

Contribution

The paper provides a simplified, elementary proof of the CLT for convex bodies, making the result more accessible and connecting thin shell properties to the CLT.

Findings

Simplified proof of Klartag's CLT for convex bodies

Demonstrates that thin shell condition implies CLT

Accessible approach using classical log-concave function facts

Abstract

We present a short proof of Klartag's central limit theorem for convex bodies, using only the most classical facts about log-concave functions. An appendix is included where we give the proof that thin shell implies CLT. The paper is accessible to anyone.

Equations

$$\underset{t\in \mathbb{R}}{\sup }\left|\mathbb{P}\left\{⟨X,\theta ⟩\le t\right\}-\Phi \left(t\right)\right|\le {\epsilon }_n$$

$$P_{E}f(x)={\int }_{E^{\perp }}f\left(x+y\right)dy$$

$$\mathbb{P}\left\{\left|\frac{\left|X\right|}{R}-1\right|<{\epsilon }^{\prime }\right\}>1-{\epsilon }^{\prime }$$

$$\hat{h}\left(\xi \right)={\int }_{{\mathbb{R}}^n}\exp \left(-2\pi i⟨\xi ,x⟩\right)h\left(x\right)dx$$

$${\hat{\varphi }}_{\sigma }(\xi )=\exp \left(-2{\pi }^2{\sigma }^2{\left|\xi \right|}^2\right)$$

$$\left|\hat{f}\left({\xi }_1\right)-\hat{f}\left({\xi }_2\right)\right|$$

$$1-\sum _{m=0}^{\infty }{\left(\frac{3}{\epsilon }\right)}^k\exp \left(-{\left\{\sqrt{c{\delta }^2+\frac{2\ln m}{n}}\right\}}^{2}n\right)\ge 1-C\exp \left(-c{\delta }^{2}n\right)$$

$$\left|\hat{f}\left(U{\left(1+\epsilon \right)}^m\sqrt{k}{\sigma }^{-1}\theta \right)-M\left({\left(1+\epsilon \right)}^m\sqrt{k}{\sigma }^{-1}\right)\right|<C\left(\delta +\sqrt{\frac{\ln m}{n}}\right){\left(1+\epsilon \right)}^m{\sigma }^{-1}\sqrt{k}$$

$$M\left(t\right)={\int }_{S^{n-1}}\hat{f}\left(t\theta \right)d{\sigma }_{n-1}\left(\theta \right)$$

$$\left|\hat{f}\left(U\xi \right)-M\left(\left|\xi \right|\right)\right|<C\left(\delta +\epsilon +\sqrt{\frac{\ln {\epsilon }^{-1}}{n}}+\sqrt{\frac{1}{n}\ln \ln \max \left\{\frac{\sigma \left|\xi \right|}{\sqrt{k}},\frac{\sqrt{k}}{\sigma \left|\xi \right|}\right\}}\right)\left|\xi \right|$$

$$\left({\mathcal{F}}_{{\mathbb{R}}^n}h\right){|}_E={\mathcal{F}}_E\left(P_{E}h\right)$$

$$P_{E}h\left(x\right)={\int }_E\exp \left(2\pi i⟨x,\xi ⟩\right)\hat{h}\left(\xi \right)d\xi$$

$$\left|P_{E}h\left(x\right)-P_{E}h\left(Wx\right)\right|$$

$$P_{E}h\left(tx\right)=e^{-A(t)}{P}_{E}h\left(ty\right)=e^{-B(t)}$$

$$A^{\prime }\left(t_x\right)t_x=k-1B^{\prime }\left(t_y\right)t_y=k-1$$

$$A\left(t_y\right)-A\left(t_x\right)$$

$$B\left(t_y\right)-B\left(t_x\right)$$

$$\underset{t\in \left\{t_x,t_y\right\}}{\sup }\left|A(t)-B(t)\right|\ge \frac{\left\{A\left(t_y\right)-A\left(t_x\right)\right\}-\left\{B\left(t_y\right)-B\left(t_x\right)\right\}}{2}=\frac{\left(k-1\right){\left(t_y-t_x\right)}^2}{2t_x{t}_y}$$

$$\left|e^{-A(t)}-e^{-B(t)}\right|$$

$$\left|e^{-A(t)}-e^{-B(t)}\right|=\left|P_{E}h\left(tx\right)-P_{E}h\left(ty\right)\right|\le C{\left(2\pi {\sigma }^2\right)}^{-(k+1)/2}\left(\delta +\sqrt{\frac{\ln n}{n}}\right)\sqrt{k}$$

$$\left|A(t)-B(t)\right|=\underset{u\in \left\{t_x,t_y\right\}}{\sup }\left|A(u)-B(u)\right|$$

$$\left|e^{-A(t)}-e^{-B(t)}\right|$$

$$\frac{t_y-t_x}{t_y}\le \gamma :=C{e}^{ck}{\left(1+{\sigma }^2\right)}^{k/4}{\sigma }^{-(k+1)/2}\left({\delta }^{1/2}+{\left(\frac{\ln n}{n}\right)}^{1/4}\right)$$

$${\int }_0^{\infty }{t}^{k-1}{P}_{E}h\left(tx\right)dt$$

$$q(t)$$

$$q\left(s\right)\le {\left[{\left(\frac{q(t)}{q\left(t_x\right)}\right)}^{1/\left(t-t_x\right)}\right]}^{s-t}q(t)\le \exp \left(-\frac{\left(k-1\right)\left(s-t\right)\left(t-t_x\right)}{3t_x^2}\right)q(t)$$

$${\int }_t^{\infty }q(s)ds\le \frac{3t_x^{2}q(t)}{\left(k-1\right)\left(t-t_x\right)}$$

$$q\left(s\right)\ge {\left[{\left(\frac{q(t_x)}{q\left(t\right)}\right)}^{1/\left(t-t_x\right)}\right]}^{t-s}q(t)\ge \exp \left(\frac{\left(k-1\right)\left(t-s\right)\left(t-t_x\right)}{3t_x^2}\right)q(t)$$

$${\int }_0^{\infty }q(s)ds\ge {\int }_{t_x}^{t}q(s)ds\ge \frac{3t_x^{2}q(t)}{\left(k-1\right)\left(t-t_x\right)}\left[\exp \left(\frac{\left(k-1\right){\left(t-t_x\right)}^2}{3t_x^2}\right)-1\right]$$

$${\int }_t^{\infty }q(s)ds\le {\left[\exp \left(\frac{\left(k-1\right){\left(t-t_x\right)}^2}{3t_x^2}\right)-1\right]}^{-1}{\int }_0^{\infty }q(s)ds$$

$${\int }_{(1-u)t_x}^{(1+u)t_x}q(s)ds\ge \left(1-C\exp \left(-ck{u}^2\right)\right)\left({\int }_0^{\infty }q(s)ds\right)$$

Full text

1 Introduction

The central limit theorem for convex bodies (Theorem 1 below) was conjectured by Brehm and Voigt [3] and independently (at about the same time) by Anttila, Ball and Perissinaki [1]. A 1998 preprint of [1] is cited in [2]. It took several years and various partial results before a full proof by Klartag emerged in [8] (see p95 for the history). A different proof was given soon afterwards by Fleury, Guédon, and Paouris [4]. Significantly improved quantitative bounds (from logarithmic to power type) were given by Klartag [9], followed by improved estimates by various authors on the related ’thin shell property’ [5, 7, 11]. More information can be found in [5, 7, 8, 9, 10, 12].

We present a simple proof that is self-contained (except for very classical results such as the Prékopa-Leindler inequality) and is accessible to anyone. The bounds on $\varepsilon_{n}$ and $\omega_{n}$ that this proof gives are poor; the contribution is simplicity. The methodology is a variation of that in Klartag’s original proof and uses Fourier inversion; the main difference being that we apply concentration directly to the Fourier transform as opposed to the measure of half-spaces. The statement of Theorem 1 below is not identical to Theorem 1.1 in [8], however under log-concavity, a uniform estimate on the cumulative distribution gives an estimate on the total variation distance, so we do indeed recover Theorem 1.1 in [8]. The standard Euclidean norm and inner product on $\mathbb{R}^{n}$ are denoted as $\left|\cdot\right|$ and $\left\langle\cdot,\cdot\right\rangle$ respectively.

Theorem 1

There exist sequences $\left(\varepsilon_{n}\right)_{1}^{\infty}$ and $\left(\omega_{n}\right)_{1}^{\infty}$ in $\left(0,\infty\right)$ with $\lim_{n\rightarrow\infty}\varepsilon_{n}=\lim_{n\rightarrow\infty}\omega_{n}=0$ such that the following is true: Let $n\in\mathbb{N}$, let $X$ be a random vector in $\mathbb{R}^{n}$ with $\mathbb{E}X=0$ and $\mathrm{Cov}\left(X\right)=I_{n}$. Assume that $X$ has a density $f=d\mu/dx$ that is log-concave, i.e. $f=e^{-g}$ where $g:\mathbb{R}^{n}\rightarrow\left(-\infty,\infty\right]$ is convex. Then there exists a set $\Theta\subset S^{n-1}$ with $\sigma_{n-1}\left(S^{n-1}\right)\geq 1-\omega_{n}$ such that for all $\theta\in\Theta$,

[TABLE]

where $\sigma_{n-1}$ is Haar measure on $S^{n-1}$ normalized so that $\sigma_{n-1}\left(S^{n-1}\right)=1$, and $\Phi\left(t\right)=\left(2\pi\right)^{-1/2}\int_{-\infty}^{t}\exp\left(-u^{2}/2\right)du$.

The proof uses two nontrivial properties of log-concave functions (see [8, 9, 10] for more details): with $f$ as in Theorem 1,

$\bullet$ If $E\subset\mathbb{R}^{n}$ is any linear subspace of dimension $1\leq k<n$, then the projection $P_{E}f:E\rightarrow\left[0,\infty\right)$ defined by

[TABLE]

is log-concave. Here integration is performed with respect to $n-k$ dimensional Lebesgue measure on $E^{\bot}$. This is a consequence of the Prékopa-Leindler inequality. Interpreting a convolution in terms of a projection of $\mathbb{R}^{n}\times\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$, we see that if $\varphi:\mathbb{R}^{n}\rightarrow\left[0,\infty\right)$ is log-concave with $\int_{\mathbb{R}^{n}}\varphi(x)dx=1$, then the convolution $f\ast\varphi$ is also log-concave.

$\bullet$ If $X$ has the thin shell property, i.e.

[TABLE]

for some $\varepsilon^{\prime},R>0$ (here we can take $R=\sqrt{n}$), then the projection of $X$ onto most one dimensional subspaces is approximately Gaussian, with estimates depending on $\varepsilon^{\prime}$. Quantitative results of this type for log-concave measures can be found in [1, 2]. For completeness, we give a precise statement with proof in Section 3.

2 Proof of Theorem 1

The proof is in three main steps.

Step 1: Approximately spherically symmetric projections. The first step mimics Milman’s proof of Dvoretzky’s theorem [14], see for example [15], but in a different way to Klartag [8, Sections 3 and 4]. Let $Y=X+\sigma Z$ for some $\sigma>0$, where $Z$ has the standard normal distribution and is independent of $X$. The density of $Y$ is $h=f\ast\phi_{\sigma}$, where $\phi_{\sigma}(x)=\left(2\pi\sigma^{2}\right)^{-n/2}\exp\left(-2^{-1}\sigma^{-2}\left|x\right|^{2}\right)$ and $\ast$ denotes convolution. Then $\widehat{h}=\widehat{f}\cdot\widehat{\phi}_{\gamma}$, where $\widehat{\cdot}$ denotes the Fourier transform,

[TABLE]

and

[TABLE]

For any $\xi_{1},\xi_{2}\in\mathbb{R}^{n}$,

[TABLE]

and we see that $\widehat{f}$ is $2\pi$-Lipschitz on $\mathbb{R}^{n}$. Let $F\in G_{n,k}$ be any fixed subspace and $U$ a random matrix uniformly distributed in $O(n)$ ($k<n$ to be determined later). Then $E=UF\in G_{n,k}$ is a random $k$-dimensional subspace uniformly distributed in $G_{n,k}$. Let $\varepsilon\in\left(0,1/2\right)$ and let $\mathcal{N}\subset S_{F}=S^{n-1}\cap F$ be an $\varepsilon$-dense subset (i.e. for all $\theta\in S_{F}$ there exists $\omega\in\mathcal{N}$ such that $\left|\theta-\omega\right|<\varepsilon$. By considering the volume of disjoint balls, such a subset can be chosen with cardinality $\left|\mathcal{N}\right|\leq\left(3/\varepsilon\right)^{k}$. Assume that $k\leq c\left(\log\varepsilon^{-1}\right)^{-1}\delta n$. By Lévy’s concentration inequality for Lipschitz functions on a sphere, see e.g. [10], and the union bound, with probability at least

[TABLE]

the following event occurs: for all $m\in\left\{0,1,2\ldots\right\}$, and all $\theta\in\mathcal{N}$,

[TABLE]

where

[TABLE]

With the same probability, the same event holds with $\left(1+\varepsilon\right)^{m}$ replaced with $\left(1+\varepsilon\right)^{-m}$. Setting $\xi^{\prime}=\left(1+\varepsilon\right)^{\pm m}\sqrt{k}\sigma^{-1}\theta$, making $m$ the subject of the formula, and using the Lipschitz property of $\widehat{f}$, with high probability, for all $\xi\in F$,

[TABLE]

Optimizing over $\varepsilon$ we set $\varepsilon=\sqrt{\left(\ln n\right)/n}$. Let $P_{E}:\mathbb{R}^{n}\rightarrow E$ denote the orthogonal projection onto $E$, let $\mathcal{F}_{\mathbb{R}^{n}}:$ $L^{1}\left(\mathbb{R}^{n}\right)\rightarrow L^{\infty}\left(\mathbb{R}^{n}\right)$ denote the Fourier transform on $\mathbb{R}^{n}$ and let $\mathcal{F}_{E}:L^{1}\left(E\right)\rightarrow L^{\infty}\left(E\right)$ denote the Fourier transform on $E$ ($E$ as a Hilbert space in its own right). Recall the definition in (1). By Fubini’s theorem, the function $P_{E}h$ is the density of the random vector $P_{E}X$ (with respect to $k$-dimensional Lebesgue measure in $E$). The Fourier transform works well with orthogonal projections, in particular

[TABLE]

where $\left(\mathcal{F}_{\mathbb{R}^{n}}f\right)|_{E}$ denotes the restriction of $\mathcal{F}_{\mathbb{R}^{n}}f$ to $E$. By Fourier inversion in $E$, for all $x\in E$,

[TABLE]

so for all $W\in O\left(E\right)$, (applying a change of variables)

[TABLE]

**Step 2: Behavior of $t\mapsto P_{E}h\left(t\theta\right)$ **(in the spirit of Lemmas 4.3 and 4.4 in [8]). Consider any $x,y\in S_{E}=E\cap S^{n-1}$ and define $A,B:\left[0,\infty\right)\rightarrow\mathbb{R}$ by

[TABLE]

Since $f$ and $\phi$ are log-concave, i.e. $-\log f$ and $-\log\phi$ are convex with values in $\left(-\infty,\infty\right]$, $h=f\ast\phi$ is also log-concave. It follows from the Prékopa-Leindler inequality (see for example the discussion in [8]) that $P_{E}h$ too is log-concave, and therefore $A$ and $B$ are convex. Since $P_{E}h=\left(P_{E}f\right)\ast\left(P_{E}\phi_{\sigma}\right)$, $A$ and $B$ are infinitely differentiable. In preparation for an integral over $E$ in polar coordinates, we now study $t\mapsto t^{k-1}e^{-A(t)}$ and $t\mapsto t^{k-1}e^{-B(t)}$, $t\in\left[0,\infty\right)$. These functions are maximized at $t_{x}$,$t_{y}\in\left(0,\infty\right)$ that satisfy

[TABLE]

Such numbers exist since $A^{\prime}(t)t$ is continuous with limit [math] (resp. $\infty$) as $t\rightarrow 0$ (resp. $t\rightarrow\infty$), similarly for $B$. After a possible re-labeling of $x$ and $y$ we may assume that $t_{x}\leq t_{y}$. Our goal is to show that these numbers cannot be too far apart (in the sense that their ratio is close to $1$). If $t_{x}=t_{y}$ there is nothing to show, so assume $t_{x}<t_{y}$. By convexity,

[TABLE]

and therefore

[TABLE]

Assume momentarily that there exists $t\in\left\{t_{x},t_{y}\right\}$ such that $A(t)-B(t)\geq 1$. Since $P_{E}h$ is the log-concave density of a random vector in $E$ with covariance $\left(1+\sigma^{2}\right)I$, it follows from Theorem 5.14 in [13] (see also (6) here) that $P_{E}h\left(0\right)\geq 2^{-7k}\left(1+\sigma^{2}\right)^{-k/2}$. By convexity again,

[TABLE]

However, by (2),

[TABLE]

We will choose the parameters $\delta$, $k$, and $\sigma$ so that the upper bound on $\left|e^{-A(t)}-e^{-B(t)}\right|$ is less than the lower bound, which implies that we may assume that $B(t)-A(t)>-1$ for all $t\in\left\{t_{x},t_{y}\right\}$. Now let $t\in\left\{t_{x},t_{y}\right\}$ such that

[TABLE]

By (3),

[TABLE]

so

[TABLE]

For an appropriate choice of parameters this will achieve our goal of showing that $t_{x}$ and $t_{y}$ cannot be too far apart (relatively). What this means is that in any direction $x\in S^{n-1}\cap E$, the function $t\mapsto t^{k-1}P_{E}h\left(tx\right)$ achieves its peak in about the same place. Our next goal is to show that the mass in

[TABLE]

is concentrated around $t_{x}$. Since $A$ lies above its tangent lines, defining $q$ by

[TABLE]

provided $\left|\frac{t}{t_{x}}-1\right|<1/2$. We now translate this to tail probabilities. Fix any $t\in\left[t_{x},3t_{x}/2\right]$ and $s\geq t$. By log-concavity of $q$,

[TABLE]

and therefore

[TABLE]

On the other hand, for any $s\in\left[t_{x},t\right]$,

[TABLE]

so

[TABLE]

and

[TABLE]

A similar bound holds for the left hand tail. Combining these,

[TABLE]

provided $u\in\left[0,1/2\right]$.

Step 3: Thin shell and small details. Now fix an arbitrary $x\in B_{2}^{n}\cap E$. By polar integration,

[TABLE]

which is the so called ’thin shell property’ of $P_{E}Y$ in $E$ (see Section 3 for more details), and by a result of Bobkov [2] (following Anttila, Ball and Perissinaki [1] in the symmetric case) this implies that with probability at least

[TABLE]

a further random projection $P_{\theta^{\prime}}P_{E}Y$ is approximately Gaussian (with mean zero and variance $1+\sigma^{2}$), where $\theta^{\prime}$ is uniformly distributed in $S_{E}$,

[TABLE]

See Theorem 2. Now $\left\langle\theta^{\prime},P_{E}Y\right\rangle=\left\langle\theta^{\prime},P_{E}X\right\rangle+\left\langle\theta^{\prime},\sigma P_{E}Z\right\rangle$, and $\left\langle\theta^{\prime},P_{E}Z\right\rangle\sim N(0,1)$. Assume that $t\geq 0$ and $\sigma\leq 1$, and consider any $\nu\in\left(0,1\right)$. Since

[TABLE]

by the union bound and (7), $\mathbb{P}\left\{\left\langle\theta^{\prime},P_{E}X\right\rangle\leq t\right\}$ is bounded below by

[TABLE]

and above by

[TABLE]

Choosing

[TABLE]

(a fairly arbitrary choice), where $c_{1}$ is chosen first to be small and then $C_{2}$ is chosen to be appropriately large, we get $\gamma\leq Cn^{-1/5}$ and the error bound reduces to

[TABLE]

the probability bound (of failure) reduces to

[TABLE]

where $C_{3}$ can be made arbitrarily large by taking $C_{2}$ large enough. The upper and lower bounds for $\left|e^{-A(t)}-e^{-B(t)}\right|$ earlier in the proof become (respectively) $Cn^{-1/2+0.1}$ and $Cn^{-0.1}$, which achieves the desired contradiction, and the required bound $k\leq c\delta^{2}\left(\ln n\right)^{-1}n$ is satisfied. Note that $P_{\theta^{\prime}}P_{E}=P_{\theta}$ where $\theta$ is uniformly distributed in $S^{n-1}$, so we have shown that the projection of $X$ onto most one dimensional subspaces is approximately Gaussian, and Theorem 1 follows.

Note: Radius of the thin shell. When stating and applying the fact that the thin shell property implies CLT, it is convenient to replace $t_{x}$ with $\sqrt{k}$ in (5). Let $W_{\theta}$ ($\theta\in S^{n-1}\cap E$) be a random variable with density proportional to $q_{\theta}(t)=t^{k-1}P_{E}h\left(t\theta\right)$, $t\geq 0$. From (4),

[TABLE]

so

[TABLE]

The last inequality follows since $\mathrm{vol}_{k-1}\left(S^{k-1}\right)\int_{S^{n-1}\cap E}\int_{0}^{\infty}q_{\theta}(s)dsd\sigma_{k-1}\left(\theta\right)=1$. Similarly,

[TABLE]

But $\mathbb{E}\left|P_{E}Y\right|^{2}=k$, so

[TABLE]

and (changing the constants involved) we may replace $t_{x}$ with $\sqrt{k}$ in (5).

Note: Lower bound on $P_{E}f(0)$. To simplify notation we work with the original function $f:\mathbb{R}^{n}\rightarrow\left[0,\infty\right)$, but the corresponding result can then be applied to $P_{E}f:E\rightarrow\left[0,\infty\right)$ by replacing $n$ with $k$. By log-concavity, $\left\{x\in\mathbb{R}^{n}:f(x)>f(0)\right\}$ is convex and there exists $\theta\in S^{n-1}$ such that $\left\langle\theta,x\right\rangle>0$ implies $f(x)\leq f(0)$. It is an interesting exercise to show that for any log-concave random variable in $\mathbb{R}$ with zero mean and unit variance, such as $\left\langle\theta,X\right\rangle$, $\mathbb{P}\left\{\left\langle\theta,X\right\rangle>0\right\}\geq\beta$ for some universal constant $\beta>0$ (actually for $\beta=e^{-1}$). Now

[TABLE]

where $\alpha_{n}$ is such that $\mathrm{vol}_{n}\left(\alpha_{n}\sqrt{\frac{n}{2\pi e}}\right)=1$ and $\alpha_{n}\rightarrow 1$ as $n\rightarrow\infty$ (and $B_{2}^{n}=\left\{x:\left|x\right|\leq 1\right\}$). Optimizing in $A$ yields

[TABLE]

In the symmetric case one gets the optimal base $\sqrt{2\pi e}$. The estimate $f(0)\geq 2^{-7n}$ can be found, for example, in [13, Theorem 5.14].

3 Appendix: Thin shell implies

CLT

For completeness we collect and prove various known results and tailor them to our specific use. We refer the reader to [2, Theorems 1.1 and 1.2, Eq. (1.7) Proposition 3.1] and [1] for a more extensive discussion. Our proof of Proposition 3.1 in [2] on the Lipschitz constant of $\theta\mapsto M\left(\theta,t\right)$ is slightly simplified.

Theorem 2

Let $\varepsilon>0$. Let $\mu$ be a probability measure on $\mathbb{R}^{k}$ with center of mass [math], identity covariance, and log-concave density $f=d\mu/dx$. If $\mu$ has the following thin shell property:

[TABLE]

then there exists $\Theta\subset S^{k-1}$ with $\sigma_{n-1}\left(\Theta\right)\geq 1-C\sqrt{k}\exp\left(-ck\varepsilon^{2}\right)$ such that for all $\theta\in\Theta$,

[TABLE]

Proof. Write $M\left(\theta,t\right)=\mu\left\{x\in\mathbb{R}^{k}:\left\langle x,\theta\right\rangle\leq t\right\}$. For any $\theta_{1},\theta_{2}\in S^{k-1}$ that are sufficiently close, say $\left|\theta_{1}-\theta_{2}\right|<1/10$,

[TABLE]

where $A\Delta B=\left(A\backslash B\right)\cup\left(B\backslash A\right)$ denotes the symmetric difference of $A$ and $B$. By projecting onto $span\left\{\theta_{1},\theta_{2}\right\}$ and identifying $span\left\{\theta_{1},\theta_{2}\right\}$ with $\mathbb{R}^{2}$, we conclude that

[TABLE]

where $q$ is the density of the measure projection of $\mu$ into $E$ (identified with $\mathbb{R}^{2}$), see (1), and $\cos\beta=\left\langle\theta_{1},\theta_{2}\right\rangle$. By the Prékopa-Leindler inequality $q$ is log-concave, and defines a probability measure with mean [math] and identity covariance. It is an elementary fact that for such a function, $q(x,y)\leq C\exp\left(-cx_{1}-cx_{2}\right)$ with universal constants $C,c>0$. By a change of variables (through translation),

[TABLE]

This implies that $M\left(\theta,t\right)$ is $Ce^{-c\left|t\right|}$-Lipschitz in $\theta$. Now let $\theta\in S^{k-1}$ be chosen randomly, uniformly distributed on $S^{k-1}$ and let $F(t)=\mathbb{E}M\left(\theta,t\right)$. By concentration on $S^{n-1}$ (see e.g. [10]) and the union bound, with probability at least $1-C\varepsilon^{-1}\exp\left(-cn\varepsilon^{2}\right)=1-C\sqrt{k}\left(k\varepsilon^{2}\right)^{-1/2}\exp\left(-ck\varepsilon^{2}\right)$, the following event occurs: for all $1\leq j\leq m$, $\left|M\left(\theta,t_{j}\right)-F(t_{j})\right|<\varepsilon$, where $m=\left\lfloor\varepsilon^{-1}\right\rfloor$ and $t_{j}=F^{-1}\left(j/m\right)$. Using monotonicity in $t$, we conclude that (with high probability) $\left|M\left(\theta,t\right)-F(t)\right|<C\varepsilon$ for all $t\in\mathbb{R}$. We now compare $F$ to $\Phi$. Let $\Phi_{k}\left(t\right)=\mathbb{P}\left\{\sqrt{k}\theta_{1}\leq t\right\}$, where $\theta$ is still uniform on $S^{k-1}$. Let $X$ be a random vector in $\mathbb{R}^{k}$ with distribution $\mu$ and independent of $\theta$. The vector $Y=\left\langle\theta,k^{1/2}\left|X\right|^{-1}X\right\rangle$ is independent of $k^{-1/2}\left|X\right|$ and has the same distribution as $\theta_{1}$. Using Fubini’s theorem and independence, and assuming $t>0$,

[TABLE]

A similar lower bound holds. For any $\delta,x>0$,

[TABLE]

It follows from rotational invariance of the standard normal distribution and uniqueness of Haar measure that if $Z$ is a standard normal vector in $\mathbb{R}^{k}$ then $\sqrt{k}\left|Z\right|^{-1}Z$ is uniformly distributed on $\sqrt{k}S^{k-1}$. Simulating $\theta=\sqrt{k}\left|Z\right|^{-1}Z$,

[TABLE]

which (after a bit of fiddling using (7) and Gaussian concentration of $\left|Z\right|$ about $k^{1/2}$) implies the well known estimate $\left|\Phi(t)-\Phi_{k}(t)\right|\leq ck^{-1/2}$ for all $t\in\mathbb{R}$ (this can also be seen by considering the density $\Phi_{k}^{\prime}$, similar details in [6, Section 3]). Putting all this together,

[TABLE]

with a similar lower bound. Similarly, this also holds for $t<0$.