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Cramér’s Theorem is Atypical

  • Nina Gantert
  • Steven Soojin Kim
  • Kavita Ramanan
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 6)

Abstract

The empirical mean of n independent and identically distributed (i.i.d.) random variables \((X_1,\dots ,X_n)\) can be viewed as a suitably normalized scalar projection of the n-dimensional random vector \(X^{(n)}\displaystyle \mathop {=}^{\cdot }\,(X_1,\dots ,X_n)\) in the direction of the unit vector \(n^{-1/2}(1,1,\dots ,1) \in \mathbb {S}^{n-1}\). The large deviation principle (LDP) for such projections as \(n\rightarrow \infty \) is given by the classical Cramér’s theorem. We prove an LDP for the sequence of normalized scalar projections of \(X^{(n)}\) in the direction of a generic unit vector \(\theta ^{(n)}\in \mathbb {S}^{n-1}\), as \(n\rightarrow \infty \). This LDP holds under fairly general conditions on the distribution of \(X_1\), and for “almost every” sequence of directions \((\theta ^{(n)})_{n\in \mathbb {N}}\). The associated rate function is “universal” in the sense that it does not depend on the particular sequence of directions. Moreover, under mild additional conditions on the law of \(X_1\), we show that the universal rate function differs from the Cramér rate function, thus showing that the sequence of directions \(n^{-1/2}(1,1,\dots ,1) \in \mathbb {S}^{n-1},\) \(n \in \mathbb {N}\), corresponding to Cramér’s theorem is atypical.

Keywords

Large deviations Projections High-dimensional product measures Cramér’s theorem Rate function 

Mathematics Subject Classification

60F10 (primary) 60D05 (secondary) 

Notes

Acknowledgments

NG and KR would like to thank ICERM, Providence, for an invitation to the program “Computational Challenges in Probability,” where some of this work was initiated. SSK and KR would also like to thank Microsoft Research New England for their hospitality during the Fall of 2014, when some of this work was completed. SSK was partially supported by a Department of Defense NDSEG fellowship. KR was partially supported by ARO grant W911NF-12-1-0222 and NSF grant DMS 1407504. The authors would like to thank an anonymous referee for helpful feedback on the exposition.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Nina Gantert
    • 1
  • Steven Soojin Kim
    • 2
  • Kavita Ramanan
    • 3
  1. 1.Faculty for MathematicsTechnical University of MunichGarchingGermany
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

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