Slow convergence of cyclic projections implies divergence of random projections and vice versa. Let $L_1,L_2,\cdots ,L_K$ be a family of $K$ closed subspaces of a Hilbert space. It is well known that although the cyclic product of the orthogonal projections on these spaces always converges in norm, random products might diverge. Moreover, in the cyclic case there is a dichotomy: the convergence is fast if and only if $L_1^{\perp }+\cdots +L_K^{\perp }$ is closed; otherwise the convergence is arbitrarily slow. We prove a parallel to this result concerning random products: we characterize those families $L_1,\cdots ,L_K$ for which all random products converge using their geometric and combinatorial structure.

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当预测的产品有差异时

周期投影的缓慢收敛意味着随机投影的发散，反之亦然。让${\mathrm{大号}}_{\mathrm{1个}}，{\mathrm{大号}}_2，\cdots ，{\mathrm{大号}}_{ķ}$ 成为一个家庭 $ķ$希尔伯特空间的封闭子空间。众所周知，尽管这些空间上正交投影的循环乘积总是在范数上收敛，但是随机乘积可能会发散。此外，在循环情况下，存在二分法：当且仅当且仅当${\mathrm{大号}}_{\mathrm{1个}}^{\perp }+\cdots +{\mathrm{大号}}_{ķ}^{\perp }$已经关了; 否则收敛会很慢。我们证明了有关随机产品的这一结果的相似之处：我们描述了这些家庭的特征${\mathrm{大号}}_{\mathrm{1个}}，\cdots ，{\mathrm{大号}}_{ķ}$ 为此，所有随机乘积都使用其几何和组合结构收敛。