There are also much simpler models, see Körner, "The Pleasures of Counting" (1996), Ch 17 ("Time and Chance").

tl;dr: if you have an expected average number of offspring κ in each generation of your fad then when κ=1 the probability that any line lasts for n generations (p_n) becomes small, yet because the average size of the nth generation is still 1, the average size of the remaining lines is p_n^-1, which becomes large.