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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