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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