UNRESTRICTED (RANDOM) PRODUCTS OF PROJECTIONS IN HILBERT SPACE: REGULARITY, ABSOLUTE CONVERGENCE OF TRAJECTORIES AND STATISTICS OF DISPLACEMENTS

Given a finite list V : = (V₁,..., V_N) of closed linear subspaces of a real Hilbert space H, let P_i denote the orthogonal projection operator onto V_i and P_i,λ := (1-λ)I+λP_i denote its relaxation with parameter λ ϵ [0, 2], i = 1,..., N. Under a mild regularity assumption on V known as "innate regularity" (which, for example, is always satisfied if each V_i has finite dimension or codimension), we show that all trajectories (x_n)₀ ^∞ resulting from the iteration x_n+1 := P_in,λ_n(x_n), where the i_n and the λ_n are completely arbitrary other than the assumption that {λ_n : N ϵ N} ⊂ [n,2-η] for some η ϵ (0,1], possess uniformly bounded displacement moments of arbitrarily small orders. In particular, we show that(formula presented) where C := C(V, η, γ) < ∞. This result strengthens prior results on norm convergence of these trajectories, known to hold under the same regularity assumption. For example, with γ = 1, it follows that the displacements series Σ(x_n+1 - x_n) converges absolutely in H. Our proof produces an effective bound on the constant C(V, η,γ), in particular, as a function of γ ϵ (0, ∞). Utilizing this bound as γ → 0, we also derive an effective bound on the distribution function on normalized displacements, and show that their decreasing rearrangement obeys a root-exponential type decay uniformly for all trajectories.