  Subordinating a multivariate L\'evy process, the subordinate, with a
univariate subordinator gives rise to a pathwise construction of a new L\'evy
process, provided the subordinator and the subordinate are independent
processes. The variance-gamma model in finance was generated accordingly from a
Brownian motion and a gamma process. Alternatively, multivariate subordination
can be used to create L\'evy processes, but this requires the subordinate to
have independent components. In this paper, we show that there exists another
operation acting on pairs $(T,X)$ of L\'evy processes which creates a L\'evy
process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary L\'evy
process with possibly dependent components. We show that this method is an
extension of both univariate and multivariate subordination and provide two
applications. We illustrate our methods giving a weak formulation of the
variance-$\alpha$-gamma process that exhibits a wider range of dependence than
using traditional subordination. Also, the variance generalised gamma
convolution class of L\'evy processes formed by subordinating Brownian motion
with Thorin subordinators is further extended using weak subordination.
