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By Samuel Moy

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Case Perhaps the most widely known version of the CLT is Theorem A (Lindeberg-Uvy). variance crZ. D. 2). We obtain Theorem B. D. random vectors with mean p and covariance matrix C. /7), -1 c” xi AN(^ t z). is n 1-1 Remark. It is not necessary, however, to assume finite variances. Feller (1966), p. 303, gives BASIC PROBABlLlTY LIMIT THEOREMS : THE CLT 29 Theorem C. D. with distributionfunction F. Then the existence ofconstants {a,,},{b,} such that i n n 1=1 XIis AN(a,, b,) holds ifand only if t2[1 - F(t) + F(-I)] ’0, U(t) t’oo, where U(t) = f-, x2 dF(x).

6. Remarks. (i) The exceptional set on which Y. fails to converge to Y is at most countably infinite. (ii) Similar theorems may be proved in terms of constructions on probmIo, However, a desirable feature ability spaces other than ([0, 11, of the present theorem is that it does permit the use of this convenient probability space. (iii) The theorem is “constructive,” not existential, as is demonstrated by the proof. 7 CONVERGENCE PROPERTIES OF TRANSFORMED SEQUENCES Given that X, + X in some sense of convergence, and given a function g, a basic question is whether g(X,) -+ g ( X ) in the same sense of convergence.

Suppose that Xn X. If G n EIXnr S ElXl’ < 00, then limn E{X:} = E{X’} and limn EIX,)’ = ElX)’. For practical implementationof the theorem, Lemma A(i), (ii), (iii) provides various sufficient conditions for uniform integrability. Justification for the trouble of verifying uniform integrability is provided by Lemma B, which shows that the uniform integrabilitycondition is essentially necessary. 4. Here we provide a collection of basic facts about it. terms of the corresponding distribution functions F, and F,and that the alternate notation Fn F is often convenient.

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An introduction to the theory of field extensions by Samuel Moy


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