Fig. 4.

Strategy of the proof of Proposition 3.8. The moment duality of ${(\tilde{N}(t),\tilde{M}(t))}_{t\ge 0}$ and ${(\tilde{X}(t),\tilde{Y}(t))}_{t\ge 0}$ is a consequence of Theorem 3.3. The laws of ${(\tilde{N}(t),\tilde{M}(t))}_{t\ge 0}$ and ${(\overline{N}(t),\overline{M}(t))}_{t\ge 0}$ agree when restricted to the reduced state-space $\{0,1\}\times {\mathbb{N}}_0$. We show the moment duality of ${(\overline{N}(t),\overline{M}(t))}_{t\ge 0}$ and ${(\overline{X}(t),\overline{Y}(t))}_{t\ge 0}$, which then allows us to conclude that the restricted laws of ${(\tilde{X}(t),\tilde{Y}(t))}_{t\ge 0}$ and ${(\overline{X}(t),\overline{Y}(t))}_{t\ge 0}$ also agree on $\{0,1\}\times [0,1]$