By A. D. Barbour, Louis H. Y. Chen

"A universal subject matter in chance idea is the approximation of complex likelihood distributions by way of easier ones, the vital restrict theorem being a classical instance. Stein's strategy is a device which makes this attainable in a large choice of occasions. conventional methods, for instance utilizing Fourier research, turn into awkward to hold via in occasions within which dependence performs an incredible half, while Stein's process can usually nonetheless be utilized to nice influence. additionally, the tactic can provide estimates for the mistake within the approximation, and never only a facts of convergence. neither is there in precept any limit at the distribution to be approximated; it may both good be general, or Poisson, or that of the complete direction of a random strategy, although the concepts have to this point been labored out in even more element for the classical approximation theorems.This quantity of lecture notes offers an in depth advent to the speculation and alertness of Stein's approach, in a kind appropriate for graduate scholars who are looking to acquaint themselves with the tactic. It contains chapters treating general, Poisson and compound Poisson approximation, approximation via Poisson procedures, and approximation by means of an arbitrary distribution, written by means of specialists within the diversified fields. The lectures take the reader from the very fundamentals of Stein's technique to the boundaries of present wisdom. ""

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2 So. 2 to get a BerryEsseen bound for the number of ones in the binary expansion of a random integer, using an exchangeable pair approach. 28 Louis H. Y. 2. Binary expansion of a random integer Let n > 2 be a natural number and X be a random variable uniformly distributed over the set {0,1, • • • , n — 1}. Let k be such that 2k~1 < n <2k. Write the binary expansion of X as k X = Y^ ^2fc~i i=l and let S = X\ + • • • + Xk be the number of ones in the binary expansion of X. When n = 2k, the distribution of S is the binomial distribution for k trials with probability 1/2, and hence can be approximated by a normal distribution.

1. 19 e ^ 2 . 19. 17. 6) + 26) if to > 6 + 5. Put MM=&(/{_&* { & | mm(S, 11,1)} = X)E(|^|[min(J, |^|) - ^ . 43. 43 ea'2 P(o < F W < 6). 44 6. 06(6 - a + 26). 4 5) < e - ° / 2 ( 5 ( 6 - a ) + 7 7 ). 1). • Before proving the main theorem, we need the following moment inequality. *

3). 5^)7. 1). • We remark that following the above lines of proof, one can prove that sup|P(W < z) - *(z)| < 7 ^ ( E C 2 / m i | > 1 } +E|^| 3 / { | £ i |< 1 } ), dispensing with the third moment assumption. We leave the proof to the reader. 1 (see Chen & Shao (2001)). 3. A lower bound Let Xi, i > l,be independent random variables with zero means and finite variances, and define Bn = X ^ i VarXj. 16) is satisfied, then the Lindeberg condition is necessary for the central limit theorem. I{B-i\x a multiple of a Lindeberg sum, since a sum of n identically distributed normal random variables is itself normally distributed, but the corresponding Lindeberg sum is not zero.