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The Davidson-Kendall problem and related results on the structure of the Wishart distribution. (English) Zbl 0694.62024
Aust. J. Stat. 30A, Spec. Issue, 272-280 (1988).
Consider a rv (U,V,W) with U,W\(\geq 0\) and \(V^ 2=UW\) a.s. R. Davidson and D. G. Kendall [see D. G. Kendall and E. F. Harding (eds.), Stochastic analysis. A tribute to the memory of Rollo Davidson (1973; Zbl 0266.60038)] have shown that if U and W are independent and nondegenerate rvs such that the left and right extremities of U are respectively 0 and infinity and the left extremity of W is 0 then the distribution of (U,V,W) is indecomposable. The author [Math. Proc. Cambridge Philos. Soc. 77, 553-558 (1975; Zbl 0314.60013)] has also studied this problem with no restrictions on left and right extremities. The main result of the present paper is:
Theorem 1. let (U,W,W) be a rv with U,W\(\geq 0\) and \(V^ 2=UW\) a.s. and the left extremity of \(U+W\) be 0. Let Q be a \(3\times 3\) real, nonsingular matrix. Then the distribution of (U,V,W)Q is decomposable iff either (i) \(U=0\) a.s. and the marginal distribution of W is decomposable or (ii) \(V=dU\) and \(W=d^ 2U\) a.s. for some real d and the marginal distribution of U is decomposable. This result completely answers a question of D. G. Kendall.
Theorem 2. Let (U,V,W) and Q be as in Theorem 1., let \(P(U=0)<1\), and let there exist no d for which \((V,W)=(dU,d^ 2U)\). Then the distribution of (U,V,W)Q is decomposable iff there exist independent rv’s (A,B) and X such that \((A^ 2,AB,B^ 2)\) is nondegenerate and for some (a,b,c) not equal to (0,0,0), the support of the distribution of X is \(\{\) (0,0,0), (a,b,c)\(\}\), \(aB^ 2+cA^ 2-2bAB=b^ 2-ac\) a.s., and \((U,V,W)=(A^ 2,AB,B^ 2)+X.\)
The following corollaries are derived. 1.) Let (U,V,W) and Q be as in Theorem 1. and additionally let \(P(U=0)<1\) and the conditional distribution of W given U be nondegenerate with positive probability. Then (U,V,W)Q is indecomposable.
2.) Let (U,V,W) be a r.v. such that U and W are independent and \(V^ 2=UW\) a.s. and \(P\{V=0\}<1\). Let Q be as in Theorem 1. Then (U,V,W)Q is decomposable iff U and W are nondegenerate with \(a^ 2U^ 2+2acU+d=0\) a.s., \(b^ 2W^ 2+2bcW+c^ 2-d=0\) a.s. and \(V=aU+bW+c\) a.s. for some reals a,b,c,d(note that \(2ab=1).\)
3.) Let X and Y be independent r.v.’s such that X is nondegenerate and \(P\{y=0\}<1\) and Q be as in Theorem 1. Then the distribution of \((X^ 2,XY,Y^ 2)Q\) is indecomposable.
Reviewer: R.Shantaram

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas