= Laziness invariance of branching extinction
{title2=$G_q(z)=qz+(1-q)G(z),\quad q<1$}
For a <Galton-Watson process>, replacing an individual's reproduction by a one-descendant uneventful step with probability $q<1$ changes its offspring <probability generating function> to the displayed mixture. Its fixed-point equation reduces to $G(z)=z$, so the smallest fixed point and hence the <extinction probability of a branching process> are unchanged. The mean is $q+(1-q)G'(1)$, which also preserves the subcritical/critical/supercritical classification. The excluded case $q=1$ never reproduces or dies and need not share the extinction probability.
Back to article page
