Name of theorem that random deviations in $n$ dimensional space tend to normal vector? I recall a theorem something roughly like the following, but I can no longer find it:

If we have a point $x$ in an $N$ dimensional space on some line $l$, and we take a random point $y$ that is $1$ unit distance away from $x$, drawn from a uniform distribution on the unit $N$-sphere around $x$. Then as $N$ goes to infinity, $y-x$ goes to the normal vector of $l$ at $x$ almost-surely (or in probability), no matter what $l$ is. 

Essentially, it says that "the higher the dimensions of a vector space, the more ways there are to move away from a curve, compared to moving along the curve" 
Does anyone know the name, or a reference, of this theorem? 
 A: This is a result of measure concentration on the sphere.
Levy's theorem tells you: Suppose $f:S^{n-1}\to\mathbb{R}$ has 
$$\lVert f\rVert_{Lip}=\sup\left\{\frac{|f(x)-f(y)|}{d(x,y)};\;x,y\in S^{n-1}\right\}\leq L,$$ 
then
$$\mathbb{P}\left(|f-\mathbb{E}f|\geq t\right)\leq 4\exp\left(-cn\frac{t^2}{L^2}\right).$$
The special case that you need is just $f(y)=y\cdot t$, where $t$ is the tangent to $l$ (Lipschitz norm 1). 
To prove this you don't need the full power of the above theorem, it is enough to use Maxwell's observation: Let $X=(X_1,\ldots,X_n)$ be a uniform vector in $S^{n-1}$, then for any $\theta\in S^{n-1}$ the random variable $X\cdot\theta$ is close in distribution to $G/\sqrt{n}$, where $G$ is a standard Gaussian.
To prove this, observe that the distribution of $X\cdot\theta$ is proportional to $(1-t^2)^{(n-3)/2}$ where $-1\leq t\leq 1$, and by Taylor approximation, when $n$ is large, it is close to $e^{-t^2n/2}$. (This is actually part of the proof of the theorem I wrote here).
To get the density function you can use the following: Assume $X\cdot \theta=X_1$ (by rotation invariance it is all the same), and note that if you fix the first coordinate to be $t$, then the other coordinates are distributed uniformly on a sphere of radius $\sqrt{1-t^2}$. Hence you can do a change of variables $[-1,1]\times S^{n-2}\to S^{n-1}$ by
$$(t,y)\mapsto (t,\sqrt{1-t^2}y).$$
Since the surface area of the lower dimensional sphere change with scaling by $(1-t^2)^{(n-2)/2}$, and the gradient of the above map adds a factor of $(1-t^2)^{-1/2}$ you get the correct density.
In recent years we got a lot of good new books about geometric functional analysis. For example:


*

*Asymptotic Geometric Analysis P1 (Artstein-Avidan, Giannopoulos, Milman).

*High Dimensional Probability (Vershynin).

*Alice and Bob Meet Banach (Aubrun, Szarek).


There are many other books, these are new and very accessible.
