I have some discrete random variable $w$ that has values (in decreasing order) $\underline{w}^\downarrow = \left( w_1, w_2, \dots , w_d \right)$ with corresponding probabilities $\underline{x}=(x_1, x_2, \dots , x_d )$. The average value is $W = \sum_i x_i w_i$, and each value $w_i$ represents a fluctuation from this average $\Delta_i = w_i - W$. I want to perform a brute-force algorithm that takes my old distribution and gives me a new one with all fluctuations bounded, i.e. $|\Delta_i |\leq c$. The algorithm is as follows \
- If $w_i - W > c$ replace it with $w_i \rightarrow W'+c$.
- If $w_j - W < -c$ replace it with $w_j \rightarrow W'-c$.
It is easy to show that this gives a c-bounded value of $W_\text{final}$ of
$$W_\text{final} = \frac{1}{X_u}\left(\sum\limits_u w_u w_u + c (X_+ - X_-) \right) $$ where $w_u$ are values of $w$ that didn't need to be bounded, $X_u = \sum_u x_u$, $X_+$ is the sum of all probabilities corresponding to $w_i$ that needed to be positively bounded (i.e. saturate $w_i = W_\text{final} + c$) and $X_-$ is the sum of all the probabilities corresponding to values $w_j$ that had to be negatively bounded (i.e. saturate $w_j = W_\text{final} - c$). This comes from solving the equation $$W_\text{final} = \sum\limits_u x_u w_u + \sum\limits_i x_i (W_\text{final}+c)+ \sum\limits_j x_j (W_\text{final}-c)$$ My question is the following: how to we choose the values $w$ to bound? We could start with the largest and smallest values of $w$ and repeatedly calculate a new average $w'$ each time, until we find one where all values are within $\pm c$ of $W_\text{final}$ but this seems very messy. The problem is very simple for the case that there are two possible values, $\underline{w} = (w_1, w_2)$, as the average fluctuation must be zero, i.e. $\Delta_1 + \Delta_2 =0$ and one fluctuation must be positive and one negative. In this algorithm we saturate our bounds, so we must bound the least probable fluctuation as $$x_1 |\Delta_1 | = x_2 |\Delta_2| $$ with $x_2 = 1-x_1$. Without loss of generality let $x_1 \geq 1/2$. If we make $|\Delta_1| = c$ then we would have $$|\Delta_2| = \frac{x_1}{1-x_1}c \geq c $$ thus giving a distribution which is not c-bounded for any value of $x$ other than $x=1/2$. So for a 2 dimensional distribution we always bound the least likely fluctuation. Can we generalise this to higher dimensions? Or is there another way of finding the correct partition of this distribution (into positively, negatively, and unbounded components)? Any advice would be greatly appreciated