  This paper deals with the existence issue of non-central Wishart
distributions which is a research topic initiated by Wishart (1928), and with
important contributions by e.g., L\'evy (1937), Gindikin (1975), Shanbhag
(1988), Peddada and Richards (1991). We present a new method involving the
theory of affine Markov processes, which reveals joint necessary conditions on
shape and non-centrality parameter. While Eaton's conjecture concerning the
necessary range of the shape parameter is confirmed, we also observe that it is
not sufficient anymore that it only belongs to the Gindikin ensemble, as is in
the central case.
