A pathwise interpretation of the Gorin-Shkolnikov identity

In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin's identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.


Introduction
Let r = {r t } 0≤t≤1 be a normalized Brownian excursion, that is, it is identical in law with a standard 3-dimensional Bessel bridge, which has the duration [0, 1], and starts from and ends at the origin; see e.g., [1,Section (2.2)] and references therein for the definition of normalized Brownian excursion and its equivalence in law with standard 3-dimensional Bessel bridge. We denote by l = {l x } x≥0 the total local time process of r; namely, by the occupation time formula, two processes r and l are related via In a recent paper [2], Gorin and Shkolnikov have found the following remarkable identity in law as a corollary to one of their results: , Corollary 2.15). The random variable X defined by is a centered Gaussian random variable with variance 1/12. * Mathematical Institute, Tohoku University, Aoba-ku, Sendai 980-8578, Japan. E-mail: hariya@math.tohoku.ac.jp In [2], they have shown that the expected value of the trace of a random operator indexed by T > 0, arising from random matrix theory, admits the representation for any T > 0; in comparison of this expression with the existing literature asserting that the expected value is equal to 2/(πT 3 ) exp (T 3 /96) for every T > 0, they have obtained Theorem 1.1 by the analytic continuation and the uniqueness of characteristic functions.
In this note, we give a proof of Theorem 1.1 without relying on random matrix theory; Jeulin's identity in law ([4, p. 264], [1, Proposition 3.6]): plays a central role in the proof.
2 Proof of Theorem 1.1 In this section, we give a proof of Theorem 1.1 and provide some relevant results.
Proof of Theorem 1.1. Recall from the representation of r by means of a stochastic differential equation (see, e.g., [5, Chapter XI, Exercise (3.11)]) that the process W = {W t } 0≤t≤1 defined by is a standard Brownian motion. We integrate both sides over [0, 1] and use Fubini's theorem on the right-hand side to see that Note that the left-hand side is a centered Gaussian random variable with variance 1/12. By Jeulin's identity (1.2), the right-hand side of (2.2) is identical in law with We change variables with t = H(x), x ≥ 0, to rewrite (2.3) as where the second line follows from the definition (1.1) of H and the third from Fubini's theorem. Combining this expression with (2.2) yields Therefore, to be more specific, the second integral in (2.4) should be written as Using the same reasoning as the above proof, we may obtain the following extension of Theorem 1.1: Proposition 2.1. For every positive integer n, the random variable has the Gaussian distribution with mean zero and variance 1/(2n + 1).
Proof. For each fixed n, we multiply both sides of (2.1) by (1 − t) n−1 and integrate them over [0, 1]. Then using Fubini's theorem, we obtain Since the left-hand side may be expressed as (1/n) 1 0 (1 − t) n dW t , we see that it is a centered Gaussian random variable with variance .
On the other hand, by Jeulin's identity (1.2), the right-hand side of (2.5) is identical in law with By (1.1), we may rewrite the integral in the last term as where we used Fubini's theorem for the second equality. Combining these leads to the conclusion.
We end this note with a comment on a relevant fact deduced from the proof of Proposition 2.1.