Distribution of largest entry in a random vector

Question

If we have a random unit vector on $\mathbb{C}^n$, drawn from the Fubini-Study metric, the marginal distribution of the squared absolute values of each of the coefficients in the vector is given by a Beta distribution with shape parameters $\alpha=1$, $\beta=n-1$. (See for example Zyczkowski and Sommers).

However, this doesn't immediately translate to a statement about the order statistics, since the variables are not independent (they are constrained to sum to 1, for example).

What is the distribution of the largest magnitude of the coefficients in the vector, that is, the first order statistic?

Answer 1

This problem was solved in Extreme statistics of complex random and quantum chaotic states (2007). The first two moments of the maximum absolute value of the elements of the random unit vector, $t=\max(|z_1|^2,|z_2|^2,\ldots |z_n|^2)$, are $$\langle t\rangle=\frac{H(n,1)}{n},\;\;\langle t^2\rangle=\frac{H^2(n,1)+H(n,2)}{n(n+1)}$$ where $H(n,k)$ is the $k$-th harmonic number. The cumulative density $F(t)$ in the large-$N$ limit is $$F(t)\rightarrow (1-e^{-nt})^n\approx\exp\left(-e^{-nt+\ln n}\right),\;\;\text{for}\;\;n\rightarrow\infty,$$ which is a Gumbel distribution.

Answer 2

$\newcommand{\C}{\mathbb C}$$\newcommand{\R}{\mathbb R}$Let $U=(U_1,\dots,U_n)$ be a random vector uniformly distributed on the unit sphere in $\C^n$. Then the distribution of the random vector $V:=(X_1,Y_1,\dots,X_n,Y_n)$ is uniform on the unit sphere in $\R^{2n}$, where $X_j:=\Re U_j$ and $Y_j:=\Im U_j$. So, $V$ equals $W:=(W_1,\dots,W_{2n})$ in distribution, where $$W_j:=\frac{Z_j}{\sqrt{Z_1^2+\dots+Z_{2n}^2}} $$ and $Z_1,\dots,Z_{2n}$ are iid standard normal random variables (r.v.'s). So, the random vector $(|U_1|^2 ,\dots,|U_n|^2)$ equals $R:=(R_1,\dots,R_n)$ in distribution, where $$R_j:=\frac{T_j}{T_1+\dots+T_n} $$ and $T_k:=Z_{2k-1}^2+Z_{2k}^2$, so that $T_1,\dots,T_n$ are iid standard exponential r.v.'s. So, $\max_j|U_j|^2$ equals $\max_j R_j$ in distribution. The distribution of $\max_j R_j$ was found a long time ago; see e.g. Irwin (1955) and historical references therein going back to as far as 1897. In particular, according to formula (1) in Irwin's paper,
\begin{equation}\label{eq:fisher} P(\max_j |U_j|^2>x)=P(\max_j R_j>x)=\sum_{j=1}^n(-1)^{j-1}\binom nj(1-jx)_+^{n-1} \end{equation} for $x\in[0,1]$.