  Solutions to linear controlled differential equations can be expressed in
terms of iterated path integrals of the driving path. This collection of
iterated integrals encodes essentially all information about the driving path.
While upper bounds for iterated path integrals are well known, lower bounds are
much less understood, and it is known only relatively recently that some type
of asymptotics for the $n$-th order iterated integral can be used to recover
some intrinsic quantitative properties of the path, such as the length of $C^1$
paths.
  In the present paper, we investigate the simplest type of rough paths (the
rough path analogue of line segments), and establish uniform upper and lower
estimates for the tail asymptotics of iterated integrals in terms of the local
variation of the path. Our methodology, which we believe is new for this
problem, involves developing paths into complex semisimple Lie algebras and
using the associated representation theory to study spectral properties of Lie
polynomials under the Lie algebraic development.
