Gaussian approximation of B-splines in Schwartz seminorms

Speaker(s)

Maciej Rzeszut

Language of the talk

English

Date

Nov. 28, 2024, 12:15 p.m.

Room

room 3160

Title in Polish

Gaussian approximation of B-splines in Schwartz seminorms

Seminar

Seminar of Probability Group

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We consider sections of the $n-1$ dimensional simplex $\Delta_{n-1}= \left\{y\in\R_+^n: \sum_k y_k= 1\right\}$ by hyperplanes $\sum x_k y_k=t$, for a vector $x$ satisfying the assumptions of Berry-Esseen theorem, i.e. $\sum x_k=0,\sum x_k^2=1$ and $m^3:=\sum\left|x_k\right|^3$ is sufficiently small. They are known to coincide with a rescaled B-spline with knots $x_1,\ldots,x_n$:
\[\mathrm{vol}_{n-2}\left\{y\in\Delta_{n-1}: \sum_{k=1}^n x_k y_k= t\right\} = (n-1)B(t):= (n-1) \sum_{k=1}^n \frac{(x_k-t)_+^{n-2}}{\prod_{j\neq k}(x_k-x_j)}.\]
We prove that $B(t/n)$ approximates the standard Gaussian density not just in the sup norm, but in any Schwartz seminorm: for any $p,q\in\mathbb{N}$ and $m\leq m_{p,q}$,
\[ \sup_{t\in\R} \left| \frac{\partial}{\partial t}\left( B(t/n)- (2\pi)^{-1/2}e^{-t^2/2} \right)\right|\leq  C_{p,q} m^3.\]
As the main tool, we develop a twofold (vector-valued and in a better norm) generalization of CLT for exponential variables.
The talk is based on joint work with M. Wojciechowski.

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