The Gorin–Shkolnikov Identity and Its Random Tree Generalization

David Clancy ()
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David Clancy: University of Washington

Journal of Theoretical Probability, 2021, vol. 34, issue 4, 2386-2420

Abstract: 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.

Keywords: Brownian excursion; Continuous state branching processes; Lévy process; Lamperti transform; Galton–Watson branching processes; Jeulin’s identity; Continuum random trees; 60F17; 60J55; 60J80 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-021-01128-y

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