The first part of this paper (Section 2) is devoted to the derivation of the asymptotic distribution of the sample proportion above a normal sample mean. The second part of the paper, leaning on the first, treats the asymptotic joint distribution of runs of various lengths above the sample mean (Section 3). The approach here proves general enough to cover asymptotic run distributions arising when the dichotomy criterion is magnitude relative to a sample function other than the sample mean and the population is other than normal.
"The Sample Mean Among the Moderate Order Statistics." Ann. Math. Statist. 33 (3) 1160 - 1166, September, 1962. https://doi.org/10.1214/aoms/1177704478