On Convergence of a family of Random Walks in the Infinite Dimensional Stiefel Manifold

In this paper, we study random walks taking values in an infinite-dimensional space — either a Hilbert space, or an infinite-dimensional manifold embedded in it, such as the Stiefel manifold. These random walks arise in problems in Shape Theory, particularly when stochastic optimization is applied. In these contexts, the random walks are defined at discrete times \(t \in \tau = \{ t_0 = 0 < t_1 < t_2 < \cdots \}\). By suitably interpolating the paths between times \(t_i\) and \(t_{i+1}\), we can view them as time-continuous random walks (for \(t \ge 0\)). A natural question arises: as the fineness of the partition \(\tau\) tends to zero, does such a family of random walks converge to a stochastic process? This paper provides some preliminary results in this direction, showing that — under appropriate conditions — weak convergence holds, in the sense of Prokhorov's theorem.