  The vector-valued $T(1)$ theorem due to Figiel, and a certain square function
estimate of Bourgain for translations of functions with a limited frequency
spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this
paper, a simplified approach to these results is presented, exploiting Nazarov,
Treil and Volberg's method of random dyadic cubes, which allows to circumvent
the most subtle parts of the original arguments.
