Convergence of Random Products of Projections Under Infinite-Periodic Selections

Let $\{P_j\}$ be a sequence of projections onto the closed subspaces ${\mathcal{M}}_j$ of a Hilbert space $\mathcal{H}$. Consider an infinite sequence $P_{i_1},P_{i_2},\dots$ with each $P_{i_n}\in \{P_1,P_2,\dots \}$, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence $\{P_{i_n}\cdots P_{i_2}{P}_{i_1}x{\}}_{n=1}^{\mathrm{\infty }}$ converge strongly or weakly to $Px$ for every $x\in \mathcal{H}$, where $P$ is the projection onto the intersection $\mathcal{M}={\bigcap }_{i=1}^{\mathrm{\infty }}{\mathcal{M}}_i$? In this paper, we present some results concerning random products of countably infinitely many projections $\{P_j{\}}_{j=1}^{\mathrm{\infty }}$ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions $\sigma \text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$, called infinite-periodic functions, and rigorously show that the sequence $\{T_{n}x\}$ defined by $$T_1:=P_{\sigma (1)}and{T}_n:=P_{\sigma (n)}{T}_{n-1}\text{for all }n\ge 2,$$ for $x\in \mathcal{H}$, converges weakly to $Px$, where $P$ is the projection onto ${\bigcap }_{j=1}^{\mathrm{\infty }}\mathcal{R}(P_j)$. We also provide some technical examples to illustrate our results.