  Let $M_n$ be the minimal position at generation $n$, of a real-valued
branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over
2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated
logarithm for the upper limits of $M_n$: upon the system's non-extinction, $
\limsup\_{n\to \infty} {1\over \log \log \log n} ( M_n - {3\over2} \log n) = 1$
almost surely. We also study the problem of moderate deviations of $M_n$:
$p(M_n- {3 \over 2} \log n > \lambda)$ for $\lambda\to \infty$ and
$\lambda=o(\log n)$. This problem is closely related to the small deviations of
a class of Mandelbrot's cascades.

Probability