  We establish an explicit, constructive approach to determine any element $X$
in the tensor algebra $\mathcal{T}\left(\mathbb{R}^d\right) =
\bigoplus_{n=0}^\infty\left(\mathbb{R}^d\right)^{\otimes n}$ from its moment
generating function. The only assumption is that $X$ has a nonzero radius of
convergence, which relaxes the condition of having an infinite radius of
convergence in the literature. The key building block of our approach is
tridiagonal antisymmetric matrices, whose sparsity offers a considerable
advantage for dimension reduction in applications. In particular, specialising
$X$ to the signature space of bounded $p$-variation paths in $\mathbb{R}^d$
with $1\leq p <2$, we show that the developments of such sparse matrices are
sufficient to separate points over the space of signatures, which yields a
refined answer to the "moment problem" concerning the signature. Based on the
above theoretical investigations, we propose a new distance function for
probability measures on the path space, termed as the "Restricted Path
Characteristic Function Distance" (RPCFD), and validate its effectiveness via
numerical experiments on hypothesis testing for examples of fractional Brownian
motions.
