Weak Limits of Parameter Estimation in Strongly Identifiable Mixture Models

We study the asymptotic behavior of global maximum likelihood estimation in finite mixtures with an unknown number of components bounded by $k$. Although convergence rates are well understood under strong identifiability, they do not determine the limiting distribution of mixing-measure loss. Under second-order strong identifiability, we establish two geometric results that connect density asymptotics to uncertainty about the mixing measure. First, we characterize the exact attainable density tangent set, accounting for the component budget shared by nonlocal vanishing components and local splitting directions. Second, we derive a sequential first-order expansion of squared Wasserstein loss, showing that its leading transportation cost is uniquely determined by the relative-density direction. Together with a Gaussian likelihood argmax theorem, these results yield joint weak convergence of the relative-density maximum likelihood estimator and $\sqrt{n}{W}_2^2({\widehat{G}}_n,G_{\ast })$. The density limit maximizes a Gaussian likelihood process over the attainable tangent set, and the induced limit of $n^{1\mathrm{/}4}{W}_2({\widehat{G}}_n,G_{\ast })$ is generally non-Gaussian. We establish consistency of an $m$-out-of-$n$ parametric bootstrap when $m\to \mathrm{\infty }$ and $m\mathrm{/}n\to 0$. Numerical studies assess the approximation and distinguish bootstrap and optimization errors; a flow-cytometry application illustrates uncertainty summaries invariant to component labeling under a working mixture model.