  We develop an approach for performing scaling analysis of $N$-step Random
Walks (RWs). The mean square end-to-end distance,
$\langle\vec{R}_{N}^{2}\rangle$, is written in terms of inner persistence
lengths (IPLs), which we define by the ensemble averages of dot products
between the walker's position and displacement vectors, at the $j$-th step. For
RW models statistically invariant under orthogonal transformations, we
analytically introduce a relation between $\langle\vec{R}_{N}^{2}\rangle$ and
the persistence length, $\lambda_{N}$, which is defined as the mean end-to-end
vector projection in the first step direction. For Self-Avoiding Walks (SAWs)
on 2D and 3D lattices we introduce a series expansion for $\lambda_{N}$, and by
Monte Carlo simulations we find that $\lambda_{\infty}$ is equal to a constant;
the scaling corrections for $\lambda_{N}$ can be second and higher order
corrections to scaling for $\langle\vec{R}_{N}^{2}\rangle$. Building SAWs with
typically one hundred steps, we estimate the exponents $\nu_{0}$ and
$\Delta_{1}$ from the IPL behavior as function of $j$. The obtained results are
in excellent agreement with those in the literature. This shows that only an
ensemble of paths with the same length is sufficient for determining the
scaling behavior of $\langle\vec{R}_{N}^{2}\rangle$, being that the whole
information needed is contained in the inner part of the paths.
