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BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out the precise convergence conclusions in the CLT,

[[math]] \frac{1}{\sqrt{n}}\sum_{i=1}^nf_i\sim g_t [[/math]]

going beyond the convergence in moments, which was established in the above.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Find an alternative proof for the moment formula

[[math]] M_k(G_t)=\sum_{\pi\in\mathcal P_2(k)}t^{|\pi|} [[/math]]

using a method of your choice.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Find a probability measure [math]\mu[/math] whose moments are given by

[[math]] M_k(\mu)=|P(k)| [[/math]]

then more generally find a measure [math]\mu_t[/math] whose moments are given by

[[math]] M_k(\mu_t)=\sum_{\pi\in P(k)}t^{|\pi|} [[/math]]

where [math]P(k)[/math] stands as usual for all partitions of [math]\{1,\ldots,k\}[/math].

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Find a probability measure [math]\nu[/math] whose moments are given by

[[math]] M_k(\nu)=|NC_2(k)| [[/math]]

then find as well a probability measure [math]\eta[/math] whose moments are given by

[[math]] M_k(\eta)=|NC(k)| [[/math]]

where [math]NC[/math] stands for “noncrossing”. Then try as well the parametric case.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Prove, with full details, that the rescaled coordinates

[[math]] y_i=\frac{x_i}{\sqrt{N}} [[/math]]

become independent with [math]N\to\infty[/math], for both the real and complex spheres.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Compute the density of the hyperspherical law at [math]N=4[/math], that is, the law of one of the coordinates over the unit sphere [math]S^3_\mathbb R\subset\mathbb R^4[/math].