Advertisement

Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1235–1272 | Cite as

On a Multivariate Strong Renewal Theorem

  • Zhiyi Chi
Article
  • 80 Downloads

Abstract

This paper takes the so-called probabilistic approach to the strong renewal theorem (SRT) for multivariate distributions in the domain of attraction of a stable law. A version of the SRT is obtained that allows any kind of lattice–nonlattice composition of a distribution. A general bound is derived to control the so-called small-n contribution, which arises from random walk paths that have a relatively small number of steps but make large cumulative moves. The asymptotic negligibility of the small-n contribution is essential to the SRT. Applications of the SRT are given, including some that provide a unified treatment to known results but with substantially weaker assumptions.

Keywords

Renewal Regular variation Infinitely divisible Large deviations 

Mathematics Subject Classification (2010)

60K05 60F10 

Notes

Acknowledgements

The author would like to thank two referees and the AE for their careful reading of the paper and useful suggestions.

References

  1. 1.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  2. 2.
    Breiman, L.: Probability. Classics in Applied Mathematics, vol. 7. Society for Industrial and Applied Mathematics, Philadelphia (1992) (corrected reprint of the 1968 original)Google Scholar
  3. 3.
    Caravenna, F.: The strong renewal theorem. arXiv:1507.07502 (2015)
  4. 4.
    Caravenna, F., Doney, R.A.: Local large deviations and the strong renewal theorem. arXiv:1612.07635 (2016)
  5. 5.
    Chi, Z.: Integral criteria for strong renewal theorems with infinite mean. Tech. Rep. 46, Department of Statistics, University of Connecticut. arXiv:1505:07622 (2014)
  6. 6.
    Chi, Z.: Strong renewal theorem with infinite mean beyond local large deviations. Ann. Appl. Probab. 25(3), 1513–1539 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Denisov, D., Dieker, A.B., Shneer, V.: Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36(5), 1946–1991 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Doney, R.A.: An analogue of the renewal theorem in higher dimensions. Proc. Lond. Math. Soc. (3) 16, 669–684 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Doney, R.A.: A bivariate local limit theorem. J. Multivar. Anal. 36(1), 95–102 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Doney, R.A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Relat. Fields 107(4), 451–465 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Doney, R.A.: The strong renewal theorem with infinite mean via local large deviations. arXiv:1507.06790 (2015)
  12. 12.
    Doney, R.A., Greenwood, P.E.: On the joint distribution of ladder variables of random walk. Probab. Theory Relat. Fields 94(4), 457–472 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Embrechts, P., Goldie, C.M.: Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure. Ann. Probab. 9(3), 468–481 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Erickson, K.B.: Strong renewal theorems with infinite mean. Trans. Am. Math. Soc. 151, 263–291 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Erickson, K.B.: A renewal theorem for distributions on \(R^1\) without expectation. Bull. Am. Math. Soc. 77, 406–410 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, Second edn. Wiley, New York (1971)zbMATHGoogle Scholar
  17. 17.
    Garsia, A., Lamperti, J.: A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37, 221–234 (1962/1963)Google Scholar
  18. 18.
    Gnedenko, B.V.: On a local limit theorem for identically distributed independent summands. Wiss. Z. Humboldt Univ. Berl. Math. Nat. Reihe 3, 287–293 (1954)MathSciNetGoogle Scholar
  19. 19.
    Greenwood, P., Omey, E., Teugels, J.L.: Harmonic renewal measures and bivariate domains of attraction in fluctuation theory. Z. Wahrsch. Verw. Geb. 61(4), 527–539 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Griffin, P.S.: Matrix normalized sums of independent identically distributed random vectors. Ann. Probab. 14(1), 224–246 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hahn, M.G., Klass, M.J.: Affine normability of partial sums of I.I.D. random vectors: a characterization. Z. Wahrsch. Verw. Geb. 69(4), 479–505 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hudson, W.N., Mason, J.D., Veeh, J.A.: The domain of normal attraction of an operator-stable law. Ann. Probab. 11(1), 178–184 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jacobson, N.: Lectures in Abstract Algebra, II: Linear Algebra. Graduate Texts in Mathematics, vol. 31. Springer, New York (1975)Google Scholar
  24. 24.
    Miroshnikov, A.L.: Bounds for the multidimensional Lévy concentration function. Theory Probab. Appl. 34(3), 535–540 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Petersen, K.: Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge (1983)CrossRefGoogle Scholar
  26. 26.
    Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Studies in Probability, vol. 4. The Clarendon Press, Oxford University Press, New York (1995)zbMATHGoogle Scholar
  27. 27.
    Resnick, S., Greenwood, P.: A bivariate stable characterization and domains of attraction. J. Multivar. Anal. 9(2), 206–221 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rvačeva, E.L.: On domains of attraction of multi-dimensional distributions. In: Selected Translations in Mathematical Statistics and Probability, vol. 2, pp. 183–205. American Mathematical Society, Providence (1962)Google Scholar
  29. 29.
    Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999) (translated from the 1990 Japanese original, revised by the author)Google Scholar
  30. 30.
    Sharpe, M.: Operator-stable probability distributions on vector groups. Trans. Am. Math. Soc. 136, 51–65 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Spitzer, F.: Principles of Random Walk. Graduate Texts in Mathematics, vol. 34, 2nd edn. Springer, New York (1976)CrossRefGoogle Scholar
  32. 32.
    Stone, C.: A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Stat. 36, 546–551 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Stone, C.: On local and ratio limit theorems. In: Le Cam, L.M., Neyman, J. (eds.) Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965/66), vol. II: Contributions to Probability Theory, Part 2, pp. 217–224. University of California Press, Berkeley (1967).Google Scholar
  34. 34.
    Tao, T., Vu, V.: Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  35. 35.
    Uchiyama, K.: Green’s functions for random walks on \({\bf Z}^N\). Proc. Lond. Math. Soc. (3) 77(1), 215–240 (1998)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Vatutin, V.A., Topchii, V.A.: A key renewal theorem for heavy tail distributions with \(\beta \in (0, 0.5]\). Theory Probab. Appl. 58(2), 387–396 (2013)MathSciNetGoogle Scholar
  37. 37.
    Williamson, J.A.: Random walks and Riesz kernels. Pac. J. Math. 25, 393–415 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zigel’, G.: Upper bounds for the concentration function in a Hilbert space. Theory Probab. Appl. 26(2), 335–349 (1981)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA

Personalised recommendations