Exercise 42, § 1

Let $n\geqslant 2$ be an integer. We denote by G the group $S_n$ endowed with its Haar measure $\mu_n$ of total mass 1. Let $\varpi : S_n\rightarrow \mathbf{R}$ be the mapping which associates to $\sigma$ the number $a+b$, where $a$ is the number of cycles in the decomposition of $\sigma$ into a product of disjoint cycles (A, I, p. 60, prop. 7) and $b$ is the number of fixed points of $\sigma$. We denote by $\nu_n$ the image measure $\nu_n=\varpi (\mu_n)$ on $\mathbf{R}$. It is a probability measure.

a) The characteristic function $\varphi_n$ of $\nu_n$ satisfies

$j=1$

for every $t\in \mathbf{R}$.

b) Let $\varpi_n$ be a Poisson law of parameter log($n$) and $\psi_n$ its characteristic function. When $n\rightarrow +\infty$, one has

uniformly for $t\in \mathbf{R}$. (Use Gauss's formula for the gamma function, cf. FVR, VII, p. 11, formula (3).)

c) Let $f_n(x) = (x-$ log($n$))$/\surd$log($n$) for $x\in \mathbf{R}$. The sequence of measures $f_n(\mu_n)$ converges tightly to the standard Gaussian law when $n\rightarrow +\infty$.