Convergence of iterative maximum-likelihood estimators

Let $V$ be the finite set of agents, let $X_v(t)$ denote the estimator held by agent $v\in V$ at iteration $t$, and let $X(\infty)$ be a common limiting estimator. Convergence conjecture. There exists a random variable $X(\infty)$ such that

$$\exists X(\infty)\;\forall v:\;X_v(t)\to X(\infty).$$

The preceding variance and covariance identities show that all agents become asymptotically identical in variance, but convergence of the estimators themselves is not established here.