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The Gorin–Shkolnikov Identity and Its Random Tree Generalization

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  • David Clancy

    (University of Washington)

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.

Suggested Citation

  • David Clancy, 2021. "The Gorin–Shkolnikov Identity and Its Random Tree Generalization," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2386-2420, December.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-021-01128-y
    DOI: 10.1007/s10959-021-01128-y
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    References listed on IDEAS

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    1. Gabriel Faraud & Stéphane Goutte, 2014. "Bessel Bridges Decomposition with Varying Dimension: Applications to Finance," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1375-1403, December.
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    3. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
    4. Duquesne, Thomas, 2009. "Continuum random trees and branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 99-129, January.
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