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We are interested in the magnitude n Sn = X1 + . . + X n = Xi . i=1 For simplicity, we shall assume that |Xi | ≤ 1 almost surely. This hypothesis can be relaxed to some control of the moments, precisely to having sub-exponential tail. Fix t > 0 and let λ > 0 be a parameter to be determined later. Our task is to estimate P (Sn > t) = P eSn > et . p By Markov inequality and using independence, we have p e−λt EeλSn = e−λt EeλXi . i Next, Taylor’s expansion and the mean zero and boundedness hypotheses can be used to show that, for every i, eλXi eλ 2 var Xi , 0 λ 1.

See also [32]. 3 (Bednorz and Latala [33]). Let x, y be random vectors in a separate Banach space (F, || · ||) such that y = ui εi for some vectors ui ∈ F i 1 and P (|ϕ (x)| P (|ϕ (y)| t) t) for all ϕ ∈ F ∗ , t > 0. Then there exists universal constant L such that: P( x t) LP ( y t/L) for all t > 0. 8 Random Vector and Jensen’s Inequality T A random vector x = (X1 , . . , Xn ) ∈ Rn is a collection of n random variables Xi on a common probability space. Its expectation is the vector T Ex = (EX1 , .

5 Norms of Matrices and Vectors See [25] for matrix norms. The matrix p-norm is defined, for 1 ≤ p ≤ ∞, as A p = max Ax x x=0 p , p 1/p n where x p |xi | = p . When p = 2, it is called spectral norm A i=1 2 = A . The Frobenius norm is defined as ⎛ A F n =⎝ n ⎞1/2 2 |aij | ⎠ , i=1 j=1 which can be computed element-wise. It is the same as the Euclidean norm on n vectors. Let C = AB. Then cij = aik bkj . Thus k=1 n AB 2 F = C 2 F n n n n 2 2 |cij | = = i=1 j=1 |aik bkj | . i=1 j=1 k=1 22 1 Mathematical Foundation n aik bkj , we find that Applying the Cauchy-Schwarz inequality to the expression k=1 AB n 2 F n n i=1 j=1 n k=1 n = i=1 k=1 = A F B a2ik a2ik n k=1 n b2kj n j=1 k=1 b2kj F.