Reinforced Galton–Watson processes I: Malthusian exponents - Bertoin, Jean; Mallein, Bastien (Random structures & algorithms, 2024)
In a reinforced Galton–Watson process with reproduction law ν$$ \boldsymbol{ν} $$ and memory parameter q∈(0,1)$$ q\in \left(0,1\right) $$, the number of children of a typical individual either, with probability q$$ q $$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability 1−q$$ 1-q $$, is given by an independent sample from ν$$ \boldsymbol{ν} $$. We estimate the average size of the population at a large generation, and in particular, we determine explicitly the Malthusian growth rate in terms of ν$$ \boldsymbol{ν} $$ and q$$ q $$. Our approach via the analysis of transport equations owes much to works by Flajolet and co‐authors.