Exercise 43, § 1

Let $k\geqslant 1$ be an integer. We denote by $\lambda$ the Lebesgue measure on $\mathbf{R}^k$. Let $(\mu_n)_{n\in\mathbf{N}}$ be a sequence of probability measures on $\mathbf{R}^k$ and let $\mu$ be a probability measure on $\mathbf{R}^k$. Suppose that the following conditions are satisfied:

(i) The Fourier transform of $\mu$ belongs to $L^1(\mathbf{R}^k)$ ;

(ii) There exists a sequence $(\sigma_n)$ in GL($k,\mathbf{R}$) such that the sequence of image measures $(\sigma_n(\mu_n))$ converges narrowly to $\mu$;

(iii) For every positive real $\tau$, let $\varphi_{n,\tau}$ be the characteristic function of the set of the $t\in \mathbf{R}^k$ such that $\|^t\sigma_n(t)\|\leqslant \tau$. One has

lim$_{a\rightarrow+\infty}$ sup$_{n\in\mathbf{N}}\int_{\|t\|\geqslant a}|\widehat{\mu}_n(^t\sigma_n(t))|\varphi_{n,\tau}(t)dt= 0$.

According to Exercise 5, condition (i) implies that the measure $\mu$ admits a continuous density $\psi$ with respect to the Lebesgue measure. We denote $\alpha =\psi (0)$. a) Let $f\in \mathscr{C}_b(\mathbf{R}^k)$ be an integrable function such that $\widehat{f}$ has compact support. One has

lim$_{n\rightarrow+\infty}|$det($1\sigma_n$)$|\int_{\mathbf{R}^k}f \mu_n=\alpha \int_{\mathbf{R}^k}f$.

(Apply the transposition formula for the Fourier transform.)

b) The formula of the preceding question is valid when $f\in \mathscr{K}(\mathbf{R}^k)$. For every measurable and bounded subset $B\subset \mathbf{R}^k$ such that the boundary of B is negligible for the Lebesgue measure, one has

lim$_{n\rightarrow+\infty}|$det($1\sigma_n$)$|\mu_n(B) =\alpha \lambda (B)$.

(For every function $f\in \mathscr{K}_{\mathbf{R}}(\mathbf{R}^k)$, and every $\varepsilon >0$, there exist integrable functions $f_1$ and $f_2$ whose Fourier transforms have compact support and such that $f_1\leqslant f\leqslant f_2$ and $\int(f_2-f_1)\leqslant \varepsilon .$)

c) Suppose that $\sigma^{-1}_n\rightarrow 0$ in the space of real matrices of type $(k, k)$ and that there exists a continuous function $\Phi :\mathbf{R}^k\rightarrow \mathbf{C}^*$ such that

uniformly on the compact subsets of $\mathbf{R}^k$. Then the conditions above are satisfied.

d) Let us take $k= 1$. Let $\mu$ be a probability measure on $\mathbf{R}$ with basis the Lebesgue measure such that the identical function $x$ of $\mathbf{R}$ belongs to $L^2(\nu )$ and such that $\mu(x) = 0$. Let $\sigma \geqslant 0$ be such that $\sigma^2=\mu(x^2)$. We suppose that $\sigma >0$. For every integer $n\geqslant 1$, let $\mu^n$ be the $n$-th convolution power of $\mu$. For every measurable bounded subset $B\subset \mathbf{R}$ whose boundary is of measure zero with respect to the Lebesgue measure, one has

lim$_{n\rightarrow+\infty}\sigma \surd \overline{n}\mu^n(B) =2\surd 1\overline{\pi}\lambda (B)$

(“fundamental local limit theorem”).