Failure of the central limit theorem for high-dimensional point-process random walks

From papers

A high-dimensional random walk on a discrete point process is a random walk on such a point process in sufficiently large dimension. Failure conjecture. There are random walks on discrete point processes in high dimensions that do not satisfy a central limit theorem. The paper states that it gives conditions ensuring a central limit theorem but no example of a random walk without one; this conjecture predicts that such examples exist.

Progress summary

Open

The conjecture remains open: no example has been found showing that a high-dimensional point-process random walk can fail the central limit theorem.

The conjecture, stated in the original paper, predicts that such failing walks exist in sufficiently high dimensions. The paper explicitly reports that no example was known there.

Known results

In dimension $d=1$ , a central limit theorem is proved.
For $d \ge 2$ , a central limit theorem is proved under the additional condition $E_P[f_e^{2+\epsilon_0}]<\infty$ for every coordinate direction $e$ .
The authors ask whether finite second moments alone suffice; no counterexample or proof of failure is recorded.

Current status (as of August 2026): The conjecture remains open; the conditional central limit theorems are established, but no high-dimensional counterexample or claimed resolution was found.

Sources

Sources & referencesView supporting material

Primary source

Noam Berger and Ron Rosenthal, “Behavior of random walk on discrete point processes”, arXiv:1110.5740 (2013).

Additional references

2 papers in this index state this conjecture (2010–2011). The statement above is taken from the most recent of them; the others are arXiv:1005.1398.

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