A characterisation of Dirichlet distributions (Q1110938)

\textit{J. N. Darroch} and \textit{D. Ratcliff} [J. Am. Stat. Assoc. 66, 641- 643 (1971; Zbl 0228.62009)] have given a characterization of the Dirichlet distributions based on the properties of independence of various functions of the random variables \((X_ 1,X_ 2,...,X_ k)\) having a joint continuous distribution over the k-dimensional simplex: \(0\leq x_ i\leq \Sigma x_ i\leq 1\). In this paper we provide a further characterization of this family of distributions essentially based on the properties of linear regression. Some extra conditions have been imposed and these are indeed indispensable.