  We show that the horoboundary of outer space for the Lipschitz metric is a
quotient of Culler and Morgan's classical boundary, two trees being identified
whenever their translation length functions are homothetic in restriction to
the set of primitive elements of $F_N$. We identify the set of Busemann points
with the set of trees with dense orbits. We also investigate a few properties
of the horoboundary of outer space for the backward Lipschitz metric, and show
in particular that it is infinite-dimensional when $N\ge 3$. We then use our
description of the horoboundary of outer space to derive an analogue of a
theorem of Furstenberg--Kifer and Hennion for random products of outer
automorphisms of $F_N$, that estimates possible growth rates of conjugacy
classes of elements of $F_N$ under such products.
