Distribution of Euclidean Distances Between Randomly Distributed Gaussian Points in n-Space

Question

I am interested in the distribution of Eucildean distance between two points in a $k$-dimensional space.

Fortunately I came across the following paper which is pretty good in deriving the CDF and PDF:

Distribution of Euclidean Distances Between Randomly Distributed Gaussian Points in n-Space

Unfortunately, I have troubles understanding some of the notations.

In particular the authors define the two points in a k-dimensional space as $\Gamma = (\gamma_1, \ldots, \gamma_k)$ and $\Psi = (\psi_1, \ldots, \psi_k)$. However, later on the CDF is derived the following: $$F(R, k) = 1 - \frac{\Gamma(\frac{k}{2}, \frac{R^2}{4})}{\Gamma(\frac{k}{2})}$$ in which $R$ seems to be defined as "the probability that the CDF is less than or equal to R". Taking aside the definition of $R$, the set $\Gamma$ seems suddenly to be a function with one or two inputs. The paper does not contain a definition of this function.

Hence my question, what does $\Gamma(x, y)$ and $\Gamma(x)$ return in this context?

Note: I tried to contact the original authors with out much success.

Answer 1

As Calvin Lin suggested in the comments,

• $\Gamma(x) = \int\limits_0^\infty t^{x-1} e^{-t}\,dt$, the Gamma function, which for integer $x$ is $\Gamma(x)=(x-1)!$
• $\Gamma(x,y) = \int\limits_y^{\infty} t^{x-1}e^{-t}\,dt$, the upper incomplete Gamma function

The work in your link looks a little excessive. It is looking at the distribution of the distance between two independent standard multivariate normal random vectors. If this is $k$-dimensional and the distance is $D$, then you could say that

• $\frac12 D^2$ has a chi-squared distribution with $k$ degrees of freedom
• $D^2$ has a gamma distribution with shape parameter $\frac k2$ and scale parameter $4$
• $\frac1{\sqrt 2} D$ has a chi distribution with $k$ degrees of freedom
• $D$ has a generalized gamma distribution with parameters $p = 2$, $d = k$, and scale parameter $2$

and the moments of these are well known

Answer 2

$F(R,k)=\mathbb{P}(\Vert X-Y\Vert\le R),$ where $X,\,Y\in\mathbb{R}^k$ and $X,Y\sim N(0,I_k).$ That is, it is the cdf of the distance between two $k$-dimensional standard gaussian random variables. Secondly, as pointed out in the comments, $\Gamma\left(\dfrac{k}{2},\dfrac{R^2}{4}\right)$ denotes the incomplete gamma function, and $\Gamma\left(\dfrac{k}{2}\right)$ is the standard gamma function.