  Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which
the number of direct descendants of one individual may be infinite with
positive probability. Assume that the standard martingale $W_n$ related to
$\mm_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a
nonnegative function which regularly varies at infinity, with exponent greater
than -1. The paper presents sufficient conditions of the almost sure
convergence of the series $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish a
criterion of finiteness of $\me W\log^+ W a(\log^+W)$ and $\me \log^+|\zi|
a(\log^+|\zi|)$, where $\zi:=Q_1+\sum_{n=2}^\infty M_1... M_n Q_{n+1}$, and
$(M_n, Q_n)$ are independent identically distributed random vectors, not
necessarily related to $\mm_n$.
