Abstract
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 >= 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.
Summary
This paper extends the theory of vague convergence from classical measure spaces to the much richer setting of random marked metric measure spaces (mmm-spaces). A mmm-space is a triple (X, d, mu) where (X, d) is a separable complete metric space and mu is a finite measure on X x E (with E a fixed Polish mark space). The Gromov-weak topology on the space X of equivalence classes of mmm-spaces is defined via convergence of polynomial functionals (monomials) that encode the metric and measure structure.
The key innovation is defining vague convergence in this setting: a sequence of locally finite measures M_n on X* = X \ {0} (where 0 is the trivial mmm-space with null measure) converges vaguely to M if M_n[F] → M[F] for all continuous bounded functionals F with support bounded away from 0. The connection to classical vague convergence is immediate: if X = [0, infinity), the map X → |X| reduces to vague convergence on (0, infinity].
The main result (Theorem 4.2, Method of Moments) shows that convergence of all moment measures M_k of order k >= 1 — combined with a Carleman-type condition ensuring the limit is determined by its moments — implies vague convergence in the Gromov-weak topology. Crucially, the k = 0 moment (the total mass, corresponding to survival probability in branching process applications) is NOT needed. If the k = 0 moment also converges, weak convergence follows.
This is particularly valuable for critical branching processes, where the many-to-few formula gives access to moments of order k >= 1 but the survival probability (k = 0 moment) requires separate and often difficult estimation. The paper also provides a continuous mapping theorem (Theorem 3.3) and an approximation/perturbation theorem (Theorem 5.1) for vague convergence.
Key Contributions
- Introduces a notion of vague convergence for measures on the space of mmm-spaces in the Gromov-weak topology
- Method of moments (Theorem 4.2): convergence of moments of order k >= 1 + Carleman condition ⇒ vague convergence
- Eliminates the need for k = 0 moment (survival probability) estimates in branching process applications
- Continuous mapping theorem for vague convergence (Theorem 3.3) — more general than existing versions
- Perturbation/approximation theorem (Theorem 5.1) for truncated mmm-spaces
- Proposition 3.1 characterises vague convergence as equivalent to weak convergence of restrictions M_n^(eps) for continuity values eps, plus convergence of total mass
- Proposition 3.2 gives asymptotic survival probability estimates from vague convergence
Methodology
The space of mmm-spaces X is equipped with the Gromov-Prohorov distance d_GP. Monomials Phi of order k are defined as integrals of bounded continuous functions phi against the k-fold product measure mu^k, encoding k-point metric/mark correlations. Vague convergence uses test functionals in C_b* (continuous bounded with support bounded away from 0). The method of moments proof (Theorem 4.2) works by: (1) introducing size-biased measures M_n^Phi that capture the law of (|X|, X-hat) where X-hat is the renormalised mmm-space; (2) using the classical method of moments to show M_n^Phi converges weakly; (3) using the Carleman condition to ensure the limit is uniquely determined; (4) applying Portmanteau-type arguments to upgrade to vague convergence. The perturbation theorem uses a complete metric D_P* for the vague topology (equation 8) defined via Laplace-transformed Prohorov distances.
Key Findings
- Vague convergence in the Gromov-weak topology is the natural framework for studying scaling limits where the total mass (population size) is random and can vanish
- For the Brownian CRT under the infinite excursion measure, vague convergence is essential since the total mass measure is infinite
- Moments of order k >= 1 suffice for vague convergence — the Carleman condition is a condition on total mass moments M_k[1] only
- Vague convergence provides a “weak” survival probability estimate (Proposition 3.2): P(|X_n| >= eps_n) ~ (1/c_n) P(|X| > 0) for suitable eps_n → 0
- The approximation theorem does not require knowing the limit in advance
Important References
- The Continuum Random Tree I — Aldous’s foundational work on the CRT, a key application of the vague convergence framework
- A note on vague convergence of measures — Basrak and Planinic’s unified vague convergence framework
- Convergence in distribution of random metric measure spaces — Greven, Pfaffelhuber, Winter’s marked Gromov-weak topology