How to get common universal vectors

We prove the existence of common universal vectors for various uncountable families of universal sequence of linear operators. In particular, we give a criterion for a one-parameter family of operators on a Banach space to have a common hypercyclic vector. This criterion relies on some tools from Probability Theory and depends on the geometry of the underlying Banach space. We also study several specific examples, such as shift operators or translation-dilation operators.