 Solution

We prove convergence of the actual sequential updates above, not just concentration of ideal equilibrium distributions. Write $M_m=\mathbb{E}{v}_m$ and $N_m=\mathbb{E}{v}_m^2$. Given $v_{m-1}$, the centered normal draw satisfiesThe conditional inverse-gamma distribution moments therefore giveFor $T_m\le n/10$, the coefficient multiplying $M_{m-1}$ is at most $1/6$, so $M_m$ stays bounded by a geometric-recursion argument. The coefficient multiplying $N_{m-1}$ is at most $1/8$, and the other terms are bounded using the first-moment bound, so $N_m$ also stays bounded. These bounds start from the finite deterministic $v_0$.

Since $T_m\to 0$, the boundedness makes every term containing $T_m{M}_{m-1}$ or $T_m^2{N}_{m-1}$ vanish. Hence $M_m\to S/n$ and $N_m\to S^2/n^2$. ConsequentlyThusThis establishes mean-square convergence for the specified Gaussian annealer. It does not assert that every implementation or cooling rule for every objective has the same guarantee, nor that mean-square convergence alone is almost-sure convergence.