Theorem: Uniform offspring-bounded path sums

For a fixed measurable label satisfying

for some finite constant $C_b$, its spine mean is finite and

Also,

Only $\mathbb{E}{\xi }^2<\infty$ is used.

Proof. It suffices to consider $C_b=1$. Equation (eq:twowalks) gives

By (eq:codingconsequences)–(eq:reversecoding),

For fixed $R$, the label $k{1}_{\{k\le R\}}$ is bounded. Subtract its conclusion in Corollary cor:boundedpath from (eq:offspringsum). With

we obtain, for each fixed $R$,

Because ${\mathbb{E}}_{\mathrm{sp}}K<\infty$, $m_R\to 0$. Height tightness consequently implies

Apply Corollary cor:boundedpath to $b{1}_{\{k\le R\}}$. The discarded absolute path sum is bounded by the displayed offspring tail, and the discarded absolute spine mean is bounded by $m_R$. First take $N\to \infty$, then $R\to \infty$. This proves (eq:pathmain). The comparison with $W$ follows from (eq:codingconsequences).