Skip to main content
SpringerLink
Log in

Moderate Deviations via Cumulants

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis, and Statulevičius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erdös–Rényi random graphs and U-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices and the number of particles in a growing box of random determinantal point processes such as the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and sine random point fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. U.S. Government Printing Office, Washington (1964). For sale by the Superintendent of Documents

    MATH  Google Scholar 

  2. Aleškevičienė, A.K.: Large deviations for U-statistics. Liet. Mat. Rink. 32(1), 7–19 (1992). doi:10.1007/BF00970968

    MathSciNet  Google Scholar 

  3. Alon, N., Spencer, J.H.: The Probabilistic Method, 3rd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken (2008). With an appendix on the life and work of Paul Erdős

    Book  MATH  Google Scholar 

  4. Amosova, N.N.: Necessity of the Cramér, Linnik and Statulevičius conditions for the probabilities of large deviations. Zap. Nauč. Semin. POMI 260(3), 9–16 (1999). doi:10.1023/A:1014543412360

    MathSciNet  Google Scholar 

  5. Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  6. Baldi, P., Rinott, Y.: Asymptotic normality of some graph-related statistics. J. Appl. Probab. 26(1), 171–175 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbour, A.D., Karoński, M., Ruciński, A.: A central limit theorem for decomposable random variables with applications to random graphs. J. Comb. Theory, Ser. B 47(2), 125–145 (1989). doi:10.1016/0095-8956(89)90014-2

    Article  MATH  Google Scholar 

  8. Chatterjee, S.: The missing log in large deviations for triangle counts. Random Struct. Algorithms 40(4), 437–451 (2012). doi:10.1002/rsa.20381

    Article  MATH  Google Scholar 

  9. Chatterjee, S., Varadhan, S.R.S.: The large deviation principle for the Erdös–Rényi random graph. Eur. J. Comb. 32(7), 1000–1017 (2011). doi:10.1016/j.ejc.2011.03.014

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, L., Goldstein, L., Shao, Q.M.: Normal Approximation by Stein’s Method, 1st edn. Probability and Its Applications. Springer, Berlin (2010)

    Google Scholar 

  11. Cramèr, H.: Sur un nouveau théorème-limite de la théorie des probabilités. Actual. Sci. Ind. 736, 5–23 (1938) (in Colloque consacré à la théorie des probabilités)

    Google Scholar 

  12. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics, vol. 38. Springer, New York (1998)

    Book  MATH  Google Scholar 

  13. Döring, H., Eichelsbacher, P.: Moderate deviations in a random graph and for the spectrum of Bernoulli random matrices. Electron. J. Probab. 14, 2636–2656 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eichelsbacher, P.: Moderate deviations for functional U-processes. Ann. Inst. Henri Poincaré Probab. Stat. 37(2), 245–273 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eichelsbacher, P., Löwe, M.: Moderate deviations for i.i.d. random variables. ESAIM Probab. Stat. 7, 209–218 (2003) (electronic). doi:10.1051/ps:2003005

    Article  MathSciNet  MATH  Google Scholar 

  16. Eichelsbacher, P., Schmock, U.: Rank-dependent moderate deviations of U-empirical measures in strong topologies. Probab. Theory Relat. Fields 126, 61–90 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eichelsbacher, P., Raič, M., Schreiber, T.: Moderate deviations for stabilizing functionals in geometric probability. Preprint, arXiv:1010.1665 (2010)

  18. Goldstein, L.: A Berry-Esseen bound with applications to counts in the Erdös–Rényi random graph. Ann. Appl. Probab. (2012, to appear). arXiv:1005.4390

  19. Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006) (electronic). doi:10.1214/154957806000000078

    Article  MathSciNet  MATH  Google Scholar 

  20. Hughes, C.P., Keating, J.P., O’Connell, N.: On the characteristic polynomial of a random unitary matrix. Commun. Math. Phys. 220(2), 429–451 (2001). doi:10.1007/s002200100453

    Article  MathSciNet  MATH  Google Scholar 

  21. Janson, S.: Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. Ann. Probab. 16(1), 305–312 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Janson, S.: Poisson approximation for large deviations. Random Struct. Algorithms 1(2), 221–229 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Janson, S., Ruciński, A.: The infamous upper tail. Random Struct. Algorithms 20(3), 317–342 (2002). Probabilistic methods in combinatorial optimization

    Article  MATH  Google Scholar 

  24. Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)

    Book  MATH  Google Scholar 

  25. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ(1/2+it). Commun. Math. Phys. 214(1), 57–89 (2000). doi:10.1007/s002200000261

    Article  MathSciNet  MATH  Google Scholar 

  26. Killip, R., Nenciu, I.C.: The unitary analogue of Jacobi matrices. Commun. Pure Appl. Math. 60(8), 1148–1188 (2007). doi:10.1002/cpa.20160

    Article  MathSciNet  MATH  Google Scholar 

  27. Lee, A.J.: U-Statistics Theory and Practice. Statistics: Textbooks and Monographs, vol. 110. Dekker, New York (1990)

    MATH  Google Scholar 

  28. Li, D., Rosalsky, A.: Precise lim sup behavior of probabilities of large deviations for sums of i.i.d. random variables. Int. J. Math. Math. Sci. 65–68, 3565–3576 (2004). doi:10.1155/S0161171204406516

    Article  MathSciNet  Google Scholar 

  29. Mehta, M.L.: Random Matrices, 3rd edn. Pure and Applied Mathematics (Amsterdam), vol. 142. Elsevier/Academic Press, Amsterdam (2004)

    MATH  Google Scholar 

  30. Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Studies in Probability, vol. 4. Clarendon/Oxford University Press, New York (1995).

    MATH  Google Scholar 

  31. Ruciński, A.: When are small subgraphs of a random graph normally distributed? Probab. Theory Relat. Fields 78(1), 1–10 (1988)

    Article  MATH  Google Scholar 

  32. Rudzkis, R., Saulis, L., Statulevičius, V.A.: A general lemma on probabilities of large deviations. Lith. Math. J. 18, 226–238 (1978). doi:10.1007/BF00972235

    Article  MATH  Google Scholar 

  33. Saulis, L., Statulevičius, V.A.: Limit Theorems for Large Deviations. Mathematics and Its Applications (Soviet Series), vol. 73. Kluwer Academic, Dordrecht (1991). Translated and revised from the 1989 Russian original

    Book  MATH  Google Scholar 

  34. Soshnikov, A.: Determinantal random point fields. Usp. Mat. Nauk 55(5), 107–160 (2000). doi: 10.1070/rm2000v055n05ABEH000321

    Article  MathSciNet  Google Scholar 

  35. Soshnikov, A.B.: Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Stat. Phys. 100(3–4), 491–522 (2000). doi:10.1023/A:1018672622921

    Article  MathSciNet  MATH  Google Scholar 

  36. Yurinsky, V.: Sums and Gaussian Vectors. Lecture Notes in Mathematics, vol. 1617. Springer, Berlin (1995)

    MATH  Google Scholar 

Download references

Acknowledgements

Both authors have been supported by Deutsche Forschungsgemeinschaft via SFB/TR 12. The first author was supported by the international research training group 1339 of the DFG.

Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 767, Technische Universität Berlin, 10623, Berlin, Germany

    Hanna Döring

  2. Fakultät für Mathematik, NA 3/68, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Peter Eichelsbacher

Authors
  1. Hanna Döring
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Eichelsbacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Eichelsbacher.

Additional information

Dedicated to the memory of Tomasz Schreiber.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Döring, H., Eichelsbacher, P. Moderate Deviations via Cumulants. J Theor Probab 26, 360–385 (2013). https://doi.org/10.1007/s10959-012-0437-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-012-0437-0

Keywords

Mathematics Subject Classification

Search

Navigation