Lemma: Path-counting identity

For every nonnegative measurable function $F$ of a path of length $h$,

where $Z_1,\dots ,Z_h$ are independent triples with law (eq:spine).

Proof. At one ancestor, summing over its selected child assigns probability $p_k{L}_k(dA)$ to each pair $(k,i)$. Independence of the offspring and marks at different nodes allows these factors to be multiplied down the ancestral path. Summing recursively gives (eq:manytoone). No independence between distinct endpoint paths is needed.