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