The Doob-McKean identity for stable L\'evy processes

We re-examine the celebrated Doob--McKean identity that identifies a conditioned one-dimensional Brownian motion as the radial part of a 3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now in the analogous setting of isotropic $\alpha$-stable processes. We find a natural analogue that matches the Brownian setting, with the role of the Brownian motion replaced by that of the isotropic $\alpha$-stable process, providing one interprets the components of the original identity in the right way.