  We develop a probabilistic approach to study the volumetric and geometric
properties of unit balls $\mathbb B_{q,1}^n$ of finite-dimensional Lorentz
sequences spaces $\ell_{q,1}^n$. More precisely, we show that the empirical
distribution of a random vector $X^{(n)}$ uniformly distributed on the volume
normalized Lorentz ball in $\mathbb R^n$ converges weakly to a compactly
supported symmetric probability distribution with explicitly given density; as
a consequence we obtain a weak Poincar\'e-Maxwell-Borel principle for any fixed
number $k\in\mathbb N$ of coordinates of $X^{(n)}$ as $n\to\infty$. Moreover,
we prove a central limit theorem for the largest coordinate of $X^{(n)}$,
demonstrating a quite different behavior than in the case of the $\ell_q^n$
balls, where a Gumbel distribution appears in the limit. Last but not least, we
prove a Schechtman-Schmuckenschl\"ager type result for the asymptotic volume of
intersections of volume normalized Lorentz and $\ell^n_p$ balls.
