Uniform Convergence of Martingales in the Branching Random Walk

In a discrete-time supercritical branching random walk, let Z⁽ⁿ⁾ be the point process formed by the nth generation. Let m(λ) be the Laplace transform of the intensity measure of Z⁽¹⁾. Then W⁽ⁿ⁾(λ) = ∫ e-λ xZ⁽ⁿ⁾(dx)/m(λ)ⁿ, which is the Laplace transform of Z⁽ⁿ⁾ normalized by its expected value, forms a martingale for any λ with |m(λ)| finite but nonzero. The convergence of these martingales uniformly in λ, for λ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit W(λ) is actually an analytic function of λ. The uniform convergence results are used to obtain extensions of known results on the growth of Z⁽ⁿ⁾(nc + D) with n, for bounded intervals D and fixed c. This forms the second part of the paper, where local large deviation results for Z⁽ⁿ⁾ which are uniform in c are considered. Finally, similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous-time models including branching Brownian motion.