Before revision:  The asymptotic frameworkis intended to approximate a small sample phenomenon: even though the total number n of observations may be large , the number of effective observations local to the cut-off is often small. Thus, while traditional asymptotic analysis in RDD requires a growing number of observations local to the cut-off as n grows, our framework allows for the number q of observations local to the cut-off to be fixed as n grows.
After revision:  The asymptotic frameworkis intended to approximate a small sample phenomenon: even though the total number n of observations may be large , the number of effective observations local to the cut-off is fixed as the sample size n diverges to infinity, while in the cut-off to be fixed as n grows.