Summary of results

Figure 2.5 illustrates our characterization of general distributions as a nested relationship between ${\cal S}^{(n)}$ and ${\cal T}^{(n)}$ for all $n \geq 2$. Observe that ${\cal S}^{(n)}$ is a proper subset of ${\cal S}^{(n+1)}$, and likewise ${\cal T}^{(n)}$ is a proper subset of ${\cal T}^{(n+1)}$ for all integers $n \geq 2$. Formally, the nested relationship between ${\cal S}^{(n)}$ and ${\cal T}^{(n)}$ is characterized in the next theorem.

Theorem 1   For all integers $n \geq 2$, ${\cal S}^{(n)}\subset {\cal T}^{(n)} \subset {\cal S}^{(n+1)}$.

The property ${\cal S}^{(n)}\subset {\cal T}^{(n)}$ is important because it will allow us to prove that the EC distribution produced by our moment matching algorithm uses a nearly minimal number of phases. The property ${\cal T}^{(n)} \subset {\cal S}^{(n+1)}$ is important in completing our characterization of ${\cal S}^{(n)}$. The latter property will follow immediately from our construction of the Complete solution. Note that it is important that the Complete solution is defined for all ${\cal PH}_3$, to prove ${\cal T}^{(n)} \subset {\cal S}^{(n+1)}$.

Figure 2.5: Characterizing ${\cal S}^{(n)}$ via ${\cal T}^{(n)}$. Solid lines delineate ${\cal S}^{(n)}$ (which is irregular) and dashed lines delineate ${\cal T}^{(n)}$ (which is regular - has a simple specification). Observe the nested structure of ${\cal S}^{(n)}$ and ${\cal T}^{(n)}$: ${\cal S}^{(n)}\subset {\cal T}^{(n)} \subset {\cal S}^{(n+1)}$ for all integers $n \geq 2$.

In [147], we also provide examples of common, practical distributions in set ${\cal S}^{(n)}$. These distributions include the Pareto distribution, the Bounded Pareto distribution (as defined in [64]), and the uniform distribution. We show conditions under which these and other distributions are in ${\cal S}^{(n)}$ for each $n \geq 2$.