Dirichlet parameters add under coarsening

The nonempty finite discrete space $S=\{1,\dots ,k\}$ is standard Borel by [F7]. Define $\alpha (B)={\sum }_{j\in B}{a}_j$ for $B\subseteq S$, equivalently $\alpha ={\sum }_{j=1}^k{a}_j{\delta }_j$. By [F7] this is a measure, with finite positive total ${\sum }_{j=1}^k{a}_j$, so [F1] applies to $(S,\mathcal{P}(S),\alpha )$. Thus the law of $P$ is that of $Z_j\mathrm{/}{\sum }_{h=1}^k{Z}_h$, where the $Z_j$ are independent $\mathrm{Gamma}(a_j,1)$ variables, and it is the Dirichlet law [F6].

For $u<1$, [F2] and the change of variables $t=(1-u)z$ in [F3] give $\mathbb{E}{e}^{u{Z}_j}=\mathrm{\Gamma }(a_j{)}^{-1}{\int }_0^{\mathrm{\infty }}{z}^{a_j-1}{e}^{-(1-u)z}dz=(1-u{)}^{-a_j}$. For $W_{\mathrm{\ell }}={\sum }_{j\in I_{\mathrm{\ell }}}{Z}_j$, independence therefore gives the joint moment-generating function ${\prod }_{\mathrm{\ell }=1}^r(1-u_{\mathrm{\ell }}{)}^{-b_{\mathrm{\ell }}}$ for $\mathrm{\mid }{u}_{\mathrm{\ell }}\mathrm{\mid }<\delta <1$, since the groups use disjoint coordinates in the finite product law [F5].

By [F5], the product law of independent $\mathrm{Gamma}(b_{\mathrm{\ell }},1)$ variables exists and has the same joint transform. Both laws have finite exponential moment at every $0<\delta <1$, since the transform at $(\delta ,\dots ,\delta )$ is finite. Applying [F4] to the difference of these two finite measures shows that $(W_1,\dots ,W_r)$ has that product Gamma law.

Normalize the grouped variables by their common positive total. The resulting vector is Dirichlet with parameters $(b_1,\dots ,b_r)$ by [F1], while $W_{\mathrm{\ell }}\mathrm{/}{\sum }_h{W}_h={\sum }_{j\in I_{\mathrm{\ell }}}{P}_j$ by step 1.1. This proves the claim for $r\ge 2$. For $r=1$ the only coordinate sum is ${\sum }_{j=1}^k{P}_j=1$, which is the stated one-category law.