Examples with a central limit theorem but infinite second moments

From papers

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.

Progress summary

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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).

Sources & referencesView supporting material

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

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