A branching process can be turned into a random walk if we only allow one individual to die and be replaced by its offspring on each step. If the offspring distributions is pk and the generating function is  then the random walk increments have P.Xi D k1/ D pk. Let Sk D 1CX1C: : :CXn and T0 D minfn W Sn D 0g.

Suppose  D P

k kpk > 1. Use Example 5.12 to show that P.T0 < 1/ D

, the solution <1 of .

/ D

.