## Arithmetic of Complex Manifolds by Barth W.P. (ed.), Lange H. (ed.) By Barth W.P. (ed.), Lange H. (ed.)

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INTEGRAL INEQUALITIES OVER A SYMMETRIC CONVEX SET that Ac^lk is symmetric about the origin and convex. Let Σ 1? Σ 2 be two positive definite covariance matrices. Xi[XSA] >ΡΣ,Σ2[ΧΕΑ]. 12) are strict. Proof. Let Xz be distributed according to Τνχθ,Σ,) for / = 1,2, and let Y be an Νφ,^ — Σ^) variable that is independent of Xj. 3. ■ Note that a special case of the above corollary occurs when both Σ! and Σ 2 are diagonal matrices. In this special case the statement is, of course, trivial. In the following we shall give a theorem that is analogous to but different from Anderson's theorem.

5. 40 3. 2. MULTIVARIATE CHI-SQUARE AND / DISTRIBUTIONS The study of inequalities for multivariate chi-square distribution was motivated mainly by the problem of constructing joint confidence regions for normal variances when the random variables are correlated. In the following we give a theorem which says that if each of the correlation matrices of the variables has the structure /, then a multivariate chi-square probability is bounded below by the product of the marginal probabilities. Again, without loss of generality the common variance may be assumed to be one.

38, 278-280. [Corrigenda (1968). Ann. Math. Statist. ] Sidâk, Z. (1967). Rectangular confidence regions for the means of multivariate normal distributions. / . Amer. Statist. Assoc. 62, 626-633. Sidàk, Z. (1968). On multivariate normal probabilities of rectangles: Their dependence on correlations. Ann. Math. Statist. 39, 1425-1434. Sidâk, Z. (1971). On probabilities of rectangles in multivariate Student distributions: Their dependence on correlations. Ann. Math. Statist. 42, 169-175. REFERENCES 35 Sidak, Z.