We count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a "local time," whose inverse is a stable subordinator of a known index, and a Brownian motion. The study of this limit process leads to some precise sample path properties of Brownian motion. These properties improve earlier results of Dvoretsky and Kahane on the existence of small oscillations in the Brownian path.
"A Conditioned Limit Theorem for Random Walk and Brownian Local Time on Square Root Boundaries." Ann. Probab. 11 (2) 227 - 261, May, 1983. https://doi.org/10.1214/aop/1176993594