Timeline for Quantitative multivariate CLT from quantitative CLT of linear combinations
Current License: CC BY-SA 4.0
6 events
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2 days ago | comment | added | Besfort | Thanks for the great reference! Perhaps it does make sense to make convergence of the characteristic function (which only depends on the linear combinations) quantitative using what I have, and then use that to get a bound on the distance between the distributions. I will try. | |
Dec 9 at 18:17 | comment | added | Liviu Nicolaescu | Up to an orthonormal change of coordinates you can assume the compenents of $Z_\infty$ are independent. For scalar random variables you can use the Fourier transform to estimate the Prohorov distance www-users.cse.umn.edu/~bobko001/papers/… | |
Dec 9 at 16:00 | comment | added | Besfort | @LiviuNicolaescu I understand that, but the emphasis is on getting quantitative bounds. Do you think the method you describe allows to transfer quantitative CLT for linear combinations to a quantitative multivariate CLT for the vector? | |
Dec 9 at 15:56 | comment | added | Liviu Nicolaescu | If you coud prove that for any $\xi=(\xi_1,\dotsc, \xi_k)$ the random variable $(\xi,Z)=\sum\xi_iZ_i$ converges in distribution to a normal random variable then, using Fourier transform you can conclude that $Z$ converges in distribution to a random vector $Z_\infty$ such that $(\xi,Z_\infty)$ is a normal random variable for any $\xi$. This forces $Z_\infty$ to be a Gaussian random vector. | |
S Dec 9 at 15:20 | review | First questions | |||
Dec 9 at 16:58 | |||||
S Dec 9 at 15:20 | history | asked | Besfort | CC BY-SA 4.0 |