By Robert J. Serfling

This paperback reprint of 1 of the easiest within the box covers a huge variety of restrict theorems helpful in mathematical facts, in addition to tools of evidence and methods of program. The manipulation of "probability" theorems to acquire "statistical" theorems is emphasised.

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**Extra resources for Approximation Theorems of Mathematical Statistics (Wiley Series in Probability and Statistics)**

**Example text**

4, respectively. 5. 11. D. 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).

Thus (L) implies (V*). Finally, check that (V*)implies Bn -, 00, n + 00. A useful special case consists of independent {Xi} with common mean p, common variance Q', and uniformly bounded vth absolute central moments, EIXi - pi's M < 00 (all i), where v > 2. A convenient multivariate extension of Theorem A is given by Rao (1973), p. 147: Theorem B. Let {XI} be independent random uectors with means {k), covariance matrices {XI} and distributionfunctions {Fl}. 3 Generalization: Double Arrays of Random Variables In the theorems previouslyconsidered,asymptotic normality was asserted for a sequence of sums XIgenerated by a single sequence X1, X2,.

An example of nonuniqueness consists of the class of density functions 4dt) = ffe-f’’4(i- a sin t”’), o < t < 00, for 0 < a < 1, all ofwhich possess the same moment sequence. For discussion of this and. other oddities, see Feller (1966), p. 224. 14 CONDITIONS FOR EXISTENCE OF MOMENTS OF A DISTRIBUTION Lemma. For any random variable X, (i) E l X l = j? P ( l X l 2 t)dt, (Sm) and (ii) if ElXl < 00, then P(lXl 2 t) = o(t-’), t + 00. PROOF. Denote by G the distribution function of ( X I and let c denote a (finite) continuity point of G.