The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good bound on the rate of convergence using the concept of the Friedrichs number has recently been shown [BGM].

A generalization of this result due to Amemiya and Ando [AA] to the product of random sequences of projection operators drawn from a fixed set also shows convergence to the projector onto the intersection subspace.

My question is: are there any known bounds on the convergence rate for the latter problem analogous to the earlier one? In my application I'm only interested in the case of finite-dimensional Hilbert spaces.

[vN] J. von Neumann, Functional operators, Annals of Mathematics Studies No. 22, Princeton University Press (1950)

[H] I. Halperin, The product of projection operators, Acta. Sci. Math. (Szeged) 23 (1962), 96-99.

[BGM] C. Badea, S. Grivaux, and V. M¨uller. A generalization of the Friedrichs angle and the method of alternating projections. Comptes Rendus Mathematique, 348(1–2):53–56, (2010).

[AA] I. Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta. Sci. Math. (Szeged) 26 (1965), 239-244.