By Robert J. Serfling

ISBN-10: 0471024031

ISBN-13: 9780471024033

This paperback reprint of 1 of the simplest within the box covers a large diversity of restrict theorems valuable in mathematical information, besides tools of facts and methods of program. The manipulation of "probability" theorems to acquire "statistical" theorems is emphasised.

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**Example text**

We are now ready to state the generalization of the Lindeberg-Evy CLT. Theorem (Donsker). D. ) and Q, as above. Then Qn =S W. ) The theorem as stated above is proved in Billingsley (19681 Section 10. However, the theorem was first established, in a different form, by Donsker (195 1). To see that the Donsker theorem contains the Lindeberg-Ltvy CLT, consider the set B, = {x(-): x(1) s a} in C[O, 13. It may be verified that B. E 1. Since we have It may be verified that B, is a W-continuityset, that is, W(dB,) = 0.

Returning to the first question, the following theorem states that the answer is "yes" if the function g is continuous with P,-probability 1. A detailed treatment covering a host of similar results may be found in Mann and Wald (1943). However, the methods of proof there are more cumbersome than the modern approaches we take here, utilizing for example the Skorokhod construction. Theorem. Suppose that g is continuous with Px-probability 1. Then (i) X, vp? 1 wp? g(X); (ii) X, 4 x =-g(X,) 3 g(X); (iii) X, S x =sg(x,) S g(x).

Then Qn =S W. ) The theorem as stated above is proved in Billingsley (19681 Section 10. However, the theorem was first established, in a different form, by Donsker (195 1). To see that the Donsker theorem contains the Lindeberg-Ltvy CLT, consider the set B, = {x(-): x(1) s a} in C[O, 13. It may be verified that B. E 1. Since we have It may be verified that B, is a W-continuityset, that is, W(dB,) = 0. 3, Donslcerâ€™s Theorem yields lim Q,(B,) = W(B,). 5(i) for discussion) that W(Ba) = @(a). Since a is chosen arbitrarily, the Lindeberg-Livy CLT follows.

### Approximation Theorems of Mathematical Statistics (Wiley Series in Probability and Statistics) by Robert J. Serfling

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