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The central limit theorem under finite second moments

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Let Theorem

denotethepaper′sCentralLimitTheoremforrandomwalksondiscretepointprocesses.The∗∗secondmoments∗∗arethemomentsofordertwoofthedistancesbetweenpointsinthecoordinatedirections.∗∗Finite−second−momentconjecture.∗∗Theoremdenote 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+ϵ0)(2+\epsilon_0)-moment for some ϵ0>0\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.

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  1. Central limit theorem under finite second moments

    Let fef_e be the distance between consecutive points in coordinate direction ee, and suppose the random walk satisfies the assumptions of Theorem

    exceptthatthemomenthypothesisisweakenedtofinitenessofthesecondmoments.∗∗Finite−second−momentconjecture.∗∗Theoremexcept 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+ϵ0)(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).

References

Primary source

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

central-limit-theoremdiscrete-point-processespoint-processesrandom-walks
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