Read e-book online American Mathematical Monthly, volume 116, number 1, january PDF

By Daniel J. Velleman

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

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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. Since the sum of independent normal variables is again normal, Wn will converge in distribution to Z, but (8) does not hold, since for all > 0 2 1(|X 1,n | ≥ )} = σ 2 E{Z 2 1(σ |Z| ≥ )} > 0.

30) When independence holds, we have the following kind of reverse implication, whose proof is deferred to Section 6. 1. Let Un and Vn , n = 1, 2, . . be two sequences of random variables such that Un and Vn are independent for every n. Then Un →d U and Un + Vn →d U implies Vn → p 0. Next, we show that the zero bias transformation enjoys the following continuity property. 2. Let Y and Yn , n = 1, 2, . . be mean zero random variables with finite, nonzero variances σ 2 = Var(Y ) and σn2 = Var(Yn ), respectively.

Similarly, block Bi−1 applies a downward force of u i on Bi at z i . Finally a downward external force of vi is applied on the left edge of Bi . The goal of the shield blocks is to aggregate the forces that should be applied on the spine blocks and to replace them by a set of fewer integral forces that are to be applied on the shield blocks. We will therefore place the shield blocks and choose the forces u i and their positions in such a way that most of the vi will be 0. ) The shield blocks are in equilibrium provided that the following balance conditions are satisfied: u i + vi + 1 = u i+1 + wi+1 , z i u i + yi vi + (yi + 12 ) = z i+1 u i+1 + xi+1 wi+1 , for 1 ≤ i ≤ k − 1.

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American Mathematical Monthly, volume 116, number 1, january 2009 by Daniel J. Velleman


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