A CENTRAL LIMIT THEOREM AND A LAW OF THE ITERATED LOGARITHM FOR THE BIGGINS MARTINGALE OF THE SUPERCRITICAL BRANCHING RANDOM WALK

作者：Iksanov Alexander; Kabluchko Zakhar

来源：Journal of Applied Probability, 2016, 53(4): 1178-1192.

DOI：10.1017/jpr.2016.73

• The Biggins martingale
• branching random walk
• central limit theorem
• law of the iterated logarithm

Let (Wn(theta))(n subset of N0) be the Biggins martingale associated with a supercritical branching random walk, and denote by W infinity(theta) its limit. Assuming essentially that the martingale (Wn(2 theta))(n is an element of N0) is uniformly integrable and that varW(1)(theta) is finite, we prove a functional central limit theorem for the tail process (W-infinity(theta) - Wn+r (theta))(r is an element of N0) and a law of the iterated logarithm for W infinity(theta) - Wn(theta) as n -> 8.

• 出版日期2016-12