Large deviations for a critical Galton–Watson process

Let $\mu(t)$ ($t=0,1,\dots$) be a Galton–Watson process with $\mu(0)=1$,
$$F(s)=\mathbf Ms^{\mu(1)},\quad F'(1)=1,\quad 0<F''(1)<\infty,\quad Q(t)=\mathbf P\{\mu(t)>0\}.$$
We prove that if $F(s)$ is an analytic function in the domain $|s|<1+\varepsilon(\varepsilon>0)$ and if for some integer $N\geqslant 2$
$$0<\frac{x}{t}\ln t\ln_{(N)}t\to\infty\qquad(t\to\infty,\,\ln_1 t=\ln t,\,\ln_{(k+1)}t=\ln_{(k)}\ln t)$$
then
$$e^x\mathbf P\{\mu(t)Q(t)>x\mid\mu(t)>0\}\to 1\qquad(t\to\infty).$$
The local limit theorem on the large deviations is proved too.