  We study the ballistic L\'evy walk stemming from an infinite mean traveling
time between collision events. Our study focuses on the density of spreading
particles all starting from a common origin, which is limited by a `light' cone
$-v_0 t<x<v_0 t$. In particular we study this density close to its maximum in
the vicinity of the `light' cone. The spreading density follows the
Lamperti-arcsine law describing typical fluctuations far from the `light' cone.
However this law blows up in the vicinity of the `light' cone horizon which is
nonphysical, in the sense that any finite time observation will never diverge.
We claim that one can find two laws for the spatial density, the first one is
the mentioned Lamperti-arcsine law describing the central part of the
distribution and the second is an infinite density illustrating the dynamics
for large $x$. We identify the relationship between a large position and the
longest traveling time describing the single big jump principle. From the
renewal theory we find that the distribution of rare events of the position is
related to the derivative of the average of the number of renewals at a short
`time' using a rate formalism.
