Vague convergence and method of moments for random metric measure spaces

We introduce a notion of vague convergence for random marked metric measure spaces. Our main result shows that convergence of the moments of order $k \ge 1$ of a random marked metric measure space is sufficient to obtain its vague convergence in the Gromov-weak topology. This result improves on previous methods of moments that also require convergence of the moment of order $k=0$, which in applications to critical branching processes amounts to estimating a survival probability. We also derive two useful companion results, namely a continuous mapping theorem and an approximation theorem for vague convergence of random marked metric measure spaces.