Limit theorem for a supercritical Galton–Watson process

Let $\mu_n$, $n=0,1,\dots$ be a Galton–Watson process, and $\tau_x+1$ the instant of first crossing of the level $x$ by the process. A limit theorem is proved for the joint distribution of the random variables
$$ \tau_x,\quad x-\mu_{\tau_x},\quad\mu_{\tau_x+1}-x\quad(x\to\infty) $$
on the assumption that $M\mu_1\ln(1+\mu_1)<\infty$.