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Galton-Watson process is a classical stochastic model for describing the evolution of a population over discrete time. In this process, every individual independently produces offspring according to a fixed distribution.
We introduce a reinforced version of the Galton-Watson process, with parameters $\nu$ and $q \in (0,1)$, such that every individual in the process reproduces as follows: with probability $1-q$, it gives birth to children according to the law $\nu$, while with probability $q$ it chooses one of its ancestors uniformly at random and gives birth to the same number of children as that ancestor.
Denoting by $Z_n$ the number of individuals alive at generation $n$ in this process, we study the asymptotic behaviour of $\mathbb E(Z_n)$, give conditions for $\mathbb{P}(Z_n \to \infty) > 0$ and describe the empirical ancestral offspring distribution of individuals at large times.
