In this article we provide new applications for exponential approximation using the framework of Peköz and Röllin (2011), which is based on Stein's method. We give error bounds for the nearly critical Galton-Watson process conditioned on non-extinction, and for the occupation times of Markov chains; for the latter, in particular, we give a new exponential approximation rate for the number of revisits to the origin for general two dimensional random walk, also known as the Erdös-Taylor theorem.
"Exponential Approximation for the Nearly Critical Galton-Watson Process and Occupation Times of Markov Chains." Electron. J. Probab. 16 1381 - 1393, 2011. https://doi.org/10.1214/EJP.v16-914