Consider a random vector $(X_1,X_2,\ldots,X_n)$ such that
1) $\; X_i\sim\text{Beta}(a_i,\sum_{j\neq i}a_j)\qquad i=1,2,\ldots,n$,
2) $\; X_1+X_2+\ldots+X_n=1$.
Can we conclude that $(X_1,X_2,\ldots,X_n)\sim\text{Dirichlet}(a_1,a_2,\ldots,a_n)$?
If the answer is no, is there another known example of joint distribution for $(X_1,X_2,\ldots,X_n)$ that satisfies conditions 1) and 2)? Thanks everyone for their help.