Examples with a central limit theorem but infinite second moments

Let the assumptions of Theorem

be the moment and structural assumptions under which the paper proves its Central Limit Theorem. A random walk on a discrete point process **satisfies a Central Limit Theorem** if it has the limiting behavior asserted there, while having all of its second moments finite means that every relevant coordinate-direction distance has finite second moment. **Infinite-second-moment examples conjecture.** There exist random walks on discrete point processes that satisfy a Central Limit Theorem but do not have all of their second moments finite. This asks whether the finite-second-moment hypothesis in Theorem

is necessary. The source explicitly presents the existence of such examples as conjectural and notes that it has not been verified.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Examples with a central limit theorem but infinite second moments
Consider random walks on discrete point processes and the moment condition in Theorem
$. **Example-existence conjecture.** There exist random walks on discrete point processes that satisfy a central limit theorem while not having all of their second moments finite. Equivalently, the condition in Theorem$
may fail to be necessary. The source explicitly presents this as a conjectured example rather than an established result, and notes that it had not been verified.
source: Ron Rosenthal, “Random walk on discrete point processes”, arXiv:1005.1398 (2011).

References

Primary source

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

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