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.