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Abstract
Let Θ(n) be a random vector uniformly distributed on the unit sphere Sn−1 in Rn. Consider the projection of the uniform distribution on the cube [−1,1]n to the line spanned by Θ(n). The projected distribution is the random probability measure μΘ(n) on R given by
μΘ(n)(A):=12n∫[−1,1]n1{⟨u,Θ(n)⟩∈A}du
for Borel subets A of R. It is well known that, with probability 1, the sequence of random probability measures μΘ(n) converges weakly to the centered Gaussian distribution with variance 1/3. We prove a large deviation principle for the sequence μΘ(n) on the space of probability measures on R with speed n. The (good) rate function is explicitly given by I(ν(α)):=−12log(1−∥α∥22) whenever ν(α) is the law of a random variable of the form
1−∥α∥22−−−−−−−√Z3√+∑k=1∞αkUk,
where Z is standard Gaussian independent of U1,U2,… which are i.i.d. Unif[−1,1], and α1≥α2≥⋯ is a nonincreasing sequence of nonnegative reals with ∥α∥2<1. We obtain a similar result for random projections of the uniform distribution on the discrete cube {−1,+1}n.
μΘ(n)(A):=12n∫[−1,1]n1{⟨u,Θ(n)⟩∈A}du
for Borel subets A of R. It is well known that, with probability 1, the sequence of random probability measures μΘ(n) converges weakly to the centered Gaussian distribution with variance 1/3. We prove a large deviation principle for the sequence μΘ(n) on the space of probability measures on R with speed n. The (good) rate function is explicitly given by I(ν(α)):=−12log(1−∥α∥22) whenever ν(α) is the law of a random variable of the form
1−∥α∥22−−−−−−−√Z3√+∑k=1∞αkUk,
where Z is standard Gaussian independent of U1,U2,… which are i.i.d. Unif[−1,1], and α1≥α2≥⋯ is a nonincreasing sequence of nonnegative reals with ∥α∥2<1. We obtain a similar result for random projections of the uniform distribution on the discrete cube {−1,+1}n.
Original language  English 

Pages (fromto)  103119 
Number of pages  17 
Journal  Studia Mathematica 
Volume  264 
Early online date  17 Dec 2021 
DOIs  
Publication status  Published  31 Dec 2022 
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 1 Finished

Random fragmentationcoalescence processes out of equilibrium
Kyprianou, A. & Rogers, T.
Engineering and Physical Sciences Research Council
30/03/20 → 31/12/22
Project: Research council