  In his, by now, classical work from 1981, Nerman made extensive use of a
crucial martingale $(W_t)_{t \geq 0}$ to prove convergence in probability, in
mean and almost surely, of supercritical general branching processes (a.k.a.
Crump-Mode-Jagers branching processes) counted with a general characteristic.
The martingale terminal value $W$ figures in the limits of his results.
  We investigate the rate at which the martingale, now called Nerman's
martingale, converges to its limit $W$. More precisely, assuming the existence
of a Malthusian parameter $\alpha > 0$ and $W_0\in L^2$, we prove a functional
central limit theorem for $(W-W_{t+s})_{s\in\mathbb{R}}$, properly normalized,
as $t\to\infty$. The weak limit is a randomly scaled time-changed Brownian
motion. Under an additional technical assumption, we prove a law of the
iterated logarithm for $W-W_t$.
