The aim of this article is to provide a simple sampling procedure to
reconstruct any monotone path from its signature. For every N, we sample a
lattice path of N steps with weights given by the coefficient of the
corresponding word in the signature. We show that these weights on lattice
paths satisfy the large deviations principle. In particular, this implies that
the probability of picking up a "wrong" path is exponentially small in N. The
argument relies on a probabilistic interpretation of the signature for monotone
paths
