Examining the evolution of the maximum of valence quark distribution weighted by Bjorken x, $h(x,t)equiv xq_V(x,t)$, we observe that $h(x,t)$ at the peak should become a one parameter function; $h(x_p,t)=Phi(x_p(t))$, where $x_p$ is the position of the peak and $t= log{Q^2}$. This observation is used to derive a new model independent relation which connects the partial derivative of the valence parton distribution functions (PDFs) in $x_p$ to the QCD evolution equation through the $x_p$-derivative of the logarithm of the function $Phi(x_p(t))$. A numerical analysis of this relation using empirical PDFs results in a observation of the exponential form of the $Phi(x_p(t)) = h(x_p,t) = Ce^{D x_p(t)}$ for leading to next-to-next leading order approximations of PDFs for the all $Q^2$ range covering four orders in magnitude. The exponent, $D$, of the observed height-position correlation function converges with the increase of the order of approximation. This result holds for all PDF sets considered. A similar relation is observed also for pion valence quark distribution, indicating that the obtained relation may be universal for any non-singlet partonic distribution. The observed height - position correlation is used also to indicate that no finite number exchanges can describe the analytic behavior of the valence quark distribution at the position of the peak at fixed $Q^2$.
Download