Invariance principle for random walks on discrete point processes

Assume the hypotheses referred to as assumptions

$$and$$

, and let Theorem

$$be the Central Limit Theorem for the random walk on a discrete point process. Under **appropriate scaling**, interpret the rescaled walk as a process-valued limit in the usual diffusive sense. **Brownian-motion conjecture.** Under assumptions$$

and

$$, Theorem$$

can be strengthened so that the random walk on a discrete point process, under appropriate scaling, converges to Brownian motion.

This is a functional strengthening of the stated Central Limit Theorem: rather than convergence only in the CLT formulation, the entire rescaled trajectory should converge to Brownian motion. The source leaves this strengthening as a conjecture.