The central limit theorem under finite second moments

Let Theorem

denote the paper's Central Limit Theorem for random walks on discrete point processes. The **second moments** are the moments of order two of the distances between points in the coordinate directions. **Finite-second-moment conjecture.** Theorem

remains true under the weaker assumption that only the second moments are finite.

The proof in the paper assumes a finite $(2+\epsilon_0)$ -moment for some $\epsilon_0>0$ , although the authors explain that finite second moments appear fundamental and that the stronger moment assumption is used for a particular estimate. Whether the stronger assumption can be removed is left open.

Equivalent formulations 1

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

Central limit theorem under finite second moments
Let $f_e$ be the distance between consecutive points in coordinate direction $e$ , and suppose the random walk satisfies the assumptions of Theorem
$except that the moment hypothesis is weakened to finiteness of the second moments. **Finite-second-moment conjecture.** Theorem$
remains true under the weak assumption that only the second moments are finite. The paper explains that the stronger $(2+\epsilon_0)$ -moment assumption was used in the proof to control an auxiliary estimate, while finite second moments are viewed as fundamental for constructing the corrector.
source: Ron Rosenthal, “Random walk on discrete point processes”, arXiv:1005.1398 (2011).

Sources & referencesView supporting material

Primary source

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

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The central limit theorem under finite second moments

Equivalent formulations 1

Central limit theorem under finite second moments

Primary source

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