A central limit theorem for convex sets by Klartag B.

By Klartag B.

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Extra info for A central limit theorem for convex sets

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Throughout this proof, α0 , C0 , C1 and Cˆ will stand for the universal constants just mentioned. We assume that inequality (24) – the main assumption of this proposition – holds, with the constant C1 as was just defined. 1, based on (24), since C0 n ≤ C1 n log n. By the conclusion of that proposition, √ √ 2 Prob{||X| − n | ≥ ε n} ≤ Ce−cε n (0 ≤ ε ≤ 1). (26) Let Z be a gaussian random vector in Rn , independent of X, with E Z = 0 and Cov(Z ) = n −α0 Id. 1(i), we know that √ Prob{|Z | ≥ 1} ≤ Prob{|Z | ≥ 20n · n 1−α0 } ≤ e−n .

The T -distance between X and Y compares only one-dimensional marginals of X and Y , hence it is weaker than the total-variation distance. The following proposition is proved by directly adapting the arguments of Naor and Romik [36]. 2. Let ε > 0, and assume that n > exp(C/ε2 ) is an integer. Suppose that X is a random vector in Rn with an isotropic, log-concave density. Let 1 ≤ k ≤ cε2 n be an integer, and let E ∈ G n,k be a random 2 k-dimensional subspace. Then, with probability greater than 1 − e−cε n of choosing E, T (Proj E (X ), Z E ) ≤ ε, where Z E is a standard gaussian random vector in the subspace E.

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