Conditional central limit theorem for critical branching random walk (Q6564545)

For a discrete time branching random walk on \(\mathbb R\) let \(Z^{(n)}(A)\) be the population size of the \(n\)-th generation in the set \(A\). The paper is concerned with the critical case \(\mathbb E[Z^{(1)}(\mathbb R)]=1\). In case the distribution of positions is standardized (in particular has finite second moment) and the offspring distibution has moments of all orders, conditional on non-extinction the authors prove convergence in distribution of \(n^{-1}Z^{(n)}(-\infty,\sqrt{n}x]\) for all \(x\in\mathbb R\). The limit distribution is uniquely determined by its moments that can be explicitly calculated by a recursion formula. The proof relies on an adaption of the many-to-few formula to the critical branching random walk, which allows to explicitly calculate the moments and to use moment convergence techniques. The method can also be applied to \(n^{-1}Z^{(n)}(-\infty,a_n x+b_n]\) in case the distribution of positions belongs to the domain of attraction of a stable distribution with normalizing constants \(a_n\) and centering constants \(b_n\).