### REGULAR VARIATION IN THE BRANCHING RANDOM WALK

**ALEKSANDER IKSANOV AND SERGEY POLOTSKIY**

*Theory of Stochastic Processes*

*Vol. 12 (28), no. 1–2, 2006, pp. 38–54*

Let $\{\mathcal{M}_n, n=0,1,\ldots\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,\ldots$, let $W_n$ be the moment generating function of $\mathcal{M}_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, $\sup_{n\geq 0}|W-W_n|$, $\sup_{n\geq 0}|W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that $\mathbb{P}\{W_1>x\}$ regularly varies at $\infty$, it is proved that $\mathbb{P}\{AW_n>x\}$ regularly varies at $\infty$ 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.

Full version