Gaussian mixture likelihood is unbounded under covariance collapse

For each fixed label pair the displayed density is continuous in $(ϵ,y_1,y_2)$ on $(0,\mathrm{\infty })\times {\mathbb{R}}^2$; the finite label space therefore makes it jointly measurable. For each observation coordinate, ${\sum }_{j=1}^2{\int }_{\mathbb{R}}\frac{1}{2}{\phi }_{a_j,s_j}(y)dy=1$ by [F1]. Lebesgue measure and finite counting measure are sigma-finite, so Tonelli [F5] gives integral $1\cdot 1=1$ for the nonnegative two-coordinate product density. This verifies the requirements of [F2] directly. Summing over the two labels in each coordinate gives the observed joint density $g_{ϵ}(y_1)g_{ϵ}(y_2)$, so its value at $(0,1)$ is $L_{ϵ}$.

At the two observations, $g_{ϵ}(0)\ge \frac{1}{2\sqrt{2\pi }ϵ}$ and $g_{ϵ}(1)\ge \frac{1}{2\sqrt{2\pi }}$. Thus $L_{ϵ}\ge 1\mathrm{/}(8\pi ϵ)$. For every $M>0$, choosing $0<ϵ<1\mathrm{/}(8\pi M)$ gives $L_{ϵ}>M$, so the likelihood is unbounded while each fixed-$ϵ$ value is finite. No finite parameter attains a global maximum.

This is compatible with [F3]–[F4]: the ascent theorem only compares consecutive admissible iterates; it neither bounds the likelihood sequence nor supplies a parameter attaining its unbounded supremum.