Sufficient and necessary conditions for decomposing the sum of random variables

Question

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on the interval $[0,1]$. My question is how to find the sufficient and necessary conditions for decomposability of $S$ into $n$ random variables $X_1,~X_2,~\cdots,X_n$ on $[0,1]$ satisfying \begin{equation} \sum_{i=1}^n h_iX_i=S,~\text{and} \end{equation} \begin{equation} \mathbb{E}\left[X_i\right]=\alpha_i. \end{equation}

Please note that the parameters $h_i$ and $\alpha_i$ are given at the start.

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Two trivial necessary conditions are given as follows:

1. $\mathbb{E}\left[S\right]=\sum_{i=1}^nh_i\alpha_i$;
2. $\mathbb{P}\left[X_1=\cdots=X_n=0|S=0\right]=1$ and $\mathbb{P}\left[X_1=\cdots=X_n=1|S=1\right]=1$.

Answer 1

As Alexandre Eremenko noted, the question is easy if you do not require independence. The necessary condition $\mathbb{E}\left[S\right]=\sum_{i=1}^nh_i\alpha_i$ noted by the OP is also sufficent (Edit: when one does not require an upper bound on the $X_i$.) If this condition holds, and $\mathbb{E}\left[S\right]>0$, then define $$X_i=\frac{\alpha_i S}{\mathbb{E}\left[S\right]}\,.$$ These variables will satisfy $\sum_i h_iX_i=S$.

Edit: If one requires an upper bound of 1 on the $X_i$, then a sufficient condition (when the $\alpha_i$ are decreasing) is to also require $$\alpha_1 \cdot \max S \le \mathbb{E}\left[S\right] \,.$$