Download American Mathematical Monthly, volume 116, number 1, january by Daniel J. Velleman PDF

By Daniel J. Velleman

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Extra info for American Mathematical Monthly, volume 116, number 1, january 2009

Example text

But now for any √ lim P(|X ∗In ,n | ≥ ) = lim P(|X ∗ | ≥ n ) = 0, n→∞ >0 n→∞ that is, X ∗In ,n → p 0. 2 show that the Lindeberg condition is sufficient for normal convergence, it is easy to see, and well known, that it is not necessary. In particular, consider the case where for all n the first summand X 1,n of Wn has the mean zero normal distribution σ Z with variance σ 2 ∈ (0, 1), and the Lindeberg condition is satisfied for the remaining variables, that is, that the limit is zero when taking the sum in (8) over all i = 1.

X i,n In particular, |Ui | ≤ 1 with probability one for all i = 2, . . , n, and therefore |X ∗In ,n | ≤ 1/σn → 0 (17) by (16). 2 may be invoked to show that the number of cycles of a random permutation is asymptotically normal. More generally, the small zero bias condition will hold for an array Xn with elements X i,n = X i /σn whenever the independent mean zero summand variables X 1 , X 2 , . . satisfy |X i | ≤ C with probability one for some constant C, and the variance σn2 of their sum Sn tends to infinity.

D(d−1)(2d−1) 3 + 1 blocks and, if balanced, Proof. The number of blocks contained in a d-stack is d 2(r − 1)2 = 1 + 1+ r =2 d(d − 1)(2d − 1) . , d2 . In preparation for proving the balance of parabolic stacks, we show in the next lemma that a slab can concentrate a set of forces acting on its top together with the weights of its own blocks down into a narrower set of forces acting on the row below it. The lemma is illustrated in Figure 16. Figure 16. A 6-slab with a grey 5-slab contained in it.

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