Exponential limit of surviving geometric branching
= Exponential limit of surviving geometric branching
{title2=$P(X_n/n>x\mid X_n>0)\to e^{-x}$}
For the stated critical geometric offspring law, the conditional upper tail is exactly $[n/(n+1)]^{\lfloor nx\rfloor}$ for $x\geq0$. Its logarithm tends to $-x$, proving an <exponential distribution> limit of rate one without a transform continuity theorem. The unconditional mass at zero tends to one, so survival conditioning changes the relevant limiting population scale.