grandes-ecoles 2021 Q43

grandes-ecoles · France · centrale-maths1__official Discrete Random Variables Convergence to a Limiting Distribution (Semicircle Law and Analogues)
Show the semicircle law in the general case:
For every function $f : \mathbb{R} \rightarrow \mathbb{R}$, continuous and bounded, $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) \xrightarrow{n \rightarrow +\infty} \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$ 
Show the semicircle law in the general case:

For every function $f : \mathbb{R} \rightarrow \mathbb{R}$, continuous and bounded,
$$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) \xrightarrow{n \rightarrow +\infty} \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$
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