Is the distribution of the maximum of 3D Bessel Bridge same as the distribution of the maximum of 1D Brownian Excursion? Is the distribution of the maximum of 3D Bessel Bridge same as the distribution of the maximum of 1D Brownian Excursion ?
Brownian excursion process: a Brownian motion that is conditioned always be positive when 0<t<1 to 0 at t=1.
three-dimensional Bessel bridge with duration 1: starts from the origin at time t = 0 and return to the origin at time t = 1.
My impression is that the distribution of the maximum of 3D Bessel Bridge is same as the distribution of the maximum of 1D Brownian Excursion, is THIS CORRECT ?
Any intuitive reason for this (or a simple derivation of this) ?
 A: Here is the intuitive reason, which can be formalized in the same way that Pitman proved his $2M-B$ Theorem (see [1] or [2], Sec. 5.5 page 140).
The 3D Bessel process $\{Z_t\}$ is a Doob $h$-transform of Brownian motion on $[0,\infty)$, with respect to the harmonic function $h(x)=x$ on $(0,\infty)$. Thus $Z_t$ can be interpreted as Brownian motion on $[0,\infty)$, conditioned to stay positive forever. Consider the graph $(t,Z_t)$ of that process and condition it to reach $\{1\}\times [0,\epsilon)$, then let $\epsilon \downarrow0$. Then the conditioned process will   converge both to the Bessel bridge and to a Brownian excursion.
Edit: The question asked by the OP was considered by Willians [3] who showed that the Bessel(3) bridge   has the same law as a Brownian excursion  (and not just their maxima). See also the discussion after figure 2 in [4].
[1]  J.W. PITMAN. One-dimensional Brownian motion and the three-dimensional Bessel process. Advances in Appl. Probability. 7, 511–526 (1975).
[2] Mörters, Peter, and Yuval Peres. Brownian motion. Vol. 30. Cambridge University Press, 2010.
https://www.yuval-peres-books.com/brownian-motion/
[3] D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc., 76:871–873, 1970.
[4] Pitman, Jim, and Marc Yor. "The law of the maximum of a Bessel bridge." Electronic Journal of Probability 4 (1999): 1-35.
