Random Operator-Valued Frames in Hilbert Spaces

Abstract

We study strongly measurable random bounded operators on separable Hilbert spaces and analyze two simple iterations driven by independent random positive contractions. The first, a Kaczmarz-like iteration, converges in mean square and almost surely and produces a random operator-valued frame. In the projection case it yields a Parseval identity. The second, a residual-weighted iteration, enjoys an exact step-by-step identity: the accumulated analysis terms plus a residual equal the identity operator. Under a mild mean-coercivity condition, the residual shrinks at a geometric rate in expectation, vanishes almost surely, and admits nonasymptotic tail bounds. As a result, the construction delivers an almost-sure Parseval frame for any independent sequence of positive contractions, not only projections.