Regular variation in the branching random walk

Abstract

initial ancestor located at the origin of the real line. For n = 0, 1, . . . , let Wn be the moment generating function of Mn normalized by its mean. Denote by AWn any of the following random variables: maximal function, square function, L1 and a.s. limit W, supn≥0 |W − Wn|, supn≥0 |Wn+1 − Wn|. Under mild moment restrictions and the assumption that {W1 > x} regularly varies at ∞, it is proved that P{AWn > x} regularly varies at ∞ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace–Stieltjes transforms. The result on the tail behaviour of W is established in two distinct ways.