UNIFORM-CONVERGENCE OF MARTINGALES IN THE BRANCHING RANDOM-WALK
成果类型：
Article
署名作者：
BIGGINS, JD
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1176989921
发表日期：
1992
页码：
137-151
关键词：
摘要：
In a discrete-time supercritical branching random walk, let Z(n) be the point process formed by the nth generation. Let m(lambda) be the Laplace transform of the intensity measure of Z(1). Then W(n)(lambda) = integral e(-lambda-x)Z(n)(dx)/m(lambda)n, which is the Laplace transform of Z(n) normalized by its expected value, forms a martingale for any lambda with \m(lambda)\ finite but nonzero. The convergence of these martingales uniformly in lambda, for lambda lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit W(lambda) is actually an analytic function of lambda. 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.