Daniel J. Velleman's American Mathematical Monthly, volume 117, number 2, PDF

By Daniel J. Velleman

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Additional info for American Mathematical Monthly, volume 117, number 2, February 2010

Example text

First we deduce a new type inequality based on Hoeffding’s [9] exponential inequality: if 1 , 2 , . . , n are independent Rademacher n random variables, a1 , a2 , . . , an are real numbers, and v 2 := i=1 ai2 , then the tail n probabilities of the random variable i=1 ai i may be bounded as follows: n P ai ≥z i ≤ 2 exp − i=1 z2 , 2v 2 z ≥ 0. (7) At the heart of these tail bounds is the following exponential moment bound: n E exp t ai i ≤ exp(t 2 v 2 /2), t ∈ R. 4. The space d ∞ is of type 2 with constant 2 log(2d).

10. For any normed space (B, · ) with finite dimension, inequality (4) is satisfied with K = dim(B). 1 with r = 1 provides an example where the constant K = dim(B) is optimal. 3. THE PROBABILISTIC APPROACH: TYPE AND COTYPE INEQUALITIES. 1. Rademacher Type and Cotype Inequalities. Let { i } denote a sequence of independent Rademacher random variables. Let 1 ≤ p < ∞. A Banach space B with norm · is said to be of (Rademacher) type p if there is a constant T p such that for all finite sequences {xi } in B, p n E n ≤ T pp i xi p xi i=1 .

D}. A first solution. Recall that for any x ∈ Rd , x r ≤ x q ≤ d 1/q−1/r x for 1 ≤ q < r ≤ ∞. r (6) Moreover, as mentioned before, n E Sn 2 2 = E X i 22 . i=1 Thus for 1 ≤ q < 2, n E Sn 2 q ≤ (d 1/q−1/2 )2 E Sn 2 2 n = d 2/q−1 E Xi 2 2 ≤ d 2/q−1 i=1 E Xi 2 q, i=1 whereas for 2 < r ≤ ∞, n E Sn 2 r ≤ E Sn 2 2 = n E Xi i=1 2 2 ≤ d 1−2/r E X i r2 . i=1 Thus we may conclude that (4) holds with K = K (d, r ) := d 2/r −1 d 1−2/r if 1 ≤ r ≤ 2, if 2 ≤ r ≤ ∞. 1 shows that this constant K (d, r ) is indeed optimal for 1 ≤ r ≤ 2.

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