Exercise 40, § 1

Let $\nu$ be a probability measure on $\mathbf{R}$ such that the identity function $x$ of $\mathbf{R}$ belongs to $L^2(\nu )$. Put $m=\nu (x)\in \mathbf{R}$ and let $\sigma \geqslant 0$ be such that $\sigma^2=\nu ((x-m)^2)$. We suppose that $\sigma >0$.

a) Let $f(t) = (t-m)/\sigma$. The image measure $\mu=f(\nu )$ belongs to $\mathscr{P}(\mathbf{R})$, it satisfies $\mu(x) = 0$ and $\mu(x^2) = 1$.

b) For $n\geqslant 1$, let $\mu^n=\mu* \cdots *\mu$ be the $n$-th convolution power of $\mu$. The sequence $(\mu^n)_{n\geqslant 1}$ converges narrowly to the probability measure of density

with respect to Lebesgue measure on $\mathbf{R}$, called the “standard Gaussian law” (“Fundamental limit theorem of probability theory”; apply the preceding exercise).

c) Let $d\geqslant 2$ be an integer number. For every integer $n\geqslant 1$, denote by $s_n$ the mapping of $[0,1]$ into $\{0, . . . , d-1\}$ which associates with $x$ the $n$-th term of the development of $x$ in base $d$, if it is unique, or the $n$-th term of the limited development of $x$ in base $d$, in the contrary case (TG, IV, p. 43, n$^o5$).

Let $\lambda$ be Lebesgue measure on $[0,1]$. Put $m_d= (d-1)/2$ and $\sigma_d=$ $\surd\overline{(d^2 - 1)/12}$. For all real numbers $a < b$, one has

lim$_{n\rightarrow+\infty}\lambda (\{x\in [0,1]|a <s_1(x) +\cdots \sigma +_d\surd s_n\overline{n}(x)-nm_d< b\})$