Large Deviations for the Empirical Distribution in the Branching Random Walk

We consider the branching random walk on the real line where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. It is well known that the normalized empirical measure converges to the Gaussian distribution for typical sets A. We therefore analyze the probability that at step n the empirical distribution differs from the Gaussian distribution by a constant \epsilon. We show that the decay is doubly exponential in either n or \sqrt{n}, depending on the set A and \epsilon, and we find the leading coefficient in the top exponent. To the best of our knowledge, this is the first time such large deviation probabilities are treated in this model.