The Gorin–Shkolnikov Identity and Its Random Tree Generalization

Abstract

In a recent pair of papers, Gorin and Shkolnikov (Ann Probab 46: 2287–2344, 2018) and Hariya (Electron Commun Probab 21: 6, 2016) have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance $$\frac{1}{12}$$ 1 12 . Lamarre and Shkolnikov generalized this to Brownian bridges (Lamarre and Shkolnikov in Ann Inst Henri Poincaré Probab Stat 55: 1402–1438, 2019) and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on n vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous results for a variety of other related models using stochastic calculus.