lines in Figure \[fj\_3c390\]. The corresponding 99.95% confidence bands for the best fitted results are also shown in Figure \[fj\_3c390\].

It is clear that the uncertainty of the measured second moment ($\sigma_{err}$) of the double-peaked broad H$\alpha$ has important effects on our final conclusion about the positive correlation for 3C390.3, because the varying range of second moment is not large, $\sim90\AA<\sigma<\sim100\AA$. If the uncertainty of the second moment is large enough to cover the varying range of the second moment, the shown positive correlation in Figure \[fj\_3c390\] should be fake. Thus, it is necessary to check whether larger uncertainties should be possible for our measured line width. Here, we simply consider the question as follows. For one given theoretical line profile of the double-peaked broad H$\alpha$ with second moment $\sigma$ (solid line shown in Figure \[3c390\]) best fitted for the observed line profile, if the corresponding uncertainty $\sigma_{err}$ for the second moment is reasonable, we will find that after being broadened by velocity $\sigma_{b}$ for the theoretical line profile, the new theoretical line profile with second moment $\sigma_{new}=\sigma+\sigma_{err}$ still well describes the observed double-peaked broad line profile of H$\alpha$. Here, the value of $\sigma_{b}$ can be determined by $\sigma_{b}^2 = 