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Given a finite collection $\mathbf{V}:=(V_1,\dots,V_N)$ of closed linearsubspaces of a real Hilbert space $H$, let $P_i$ denote the orthogonalprojection operator onto $V_i$ and $P_{i,\lambda}:= (1-\lambda)I + \lambda P_i$denote its relaxation with parameter $\lambda \in [0,2]$, $i=1,\dots,N$. Undera mild regularity assumption on $\mathbf{V}$ known as `innate regularity'(which, for example, is always satisfied if each $V_i$ has finite dimension orcodimension), we show that all trajectories $(x_n)_{0}^\infty$ resulting fromthe iteration $x_{n+1} := P_{i_n,\lambda_n}(x_n)$, where the $i_n$ and the$\lambda_n$ are unrestricted other than the assumption that $\{\lambda_n : n\in \mathbb{N}\} \subset [\eta,2{-}\eta]$ for some $\eta \in (0,1]$, possessuniformly bounded displacement moments of arbitrarily small orders. Inparticular, we show that $$ \sum_{n=0}^\infty \|x_{n+1} - x_n \|^\gamma \leq C\|x_0\|^\gamma ~\mbox{ for all }~ \gamma > 0,$$ where$C:=C(\mathbf{V},\eta,\gamma)<\infty$. This result strengthens prior results onnorm convergence of these trajectories, known to hold under the same regularityassumption. For example, with $\gamma=1$, it follows that the displacementsseries $\sum (x_{n+1}-x_n)$ converges absolutely in $H$.  Quantifying the constant $C(\mathbf{V},\eta,\gamma)$, we also derive aneffective bound on the distribution function of the norms of the displacements(normalized by the norm of the initial condition) which yields aroot-exponential type decay bound on their decreasing rearrangement, againuniformly for all trajectories.

Unrestricted iterations of relaxed projections in Hilbert space: Regularity, absolute convergence, and statistics of displacements