Electronic Journal of Probability

Local limit theorems and renewal theory with no moments

Kenneth S. Alexander and Quentin Berger

Full-text: Open access

Abstract

We study i.i.d. sums $\tau _k$ of nonnegative variables with index $0$: this means ${\mathbf P}(\tau _1=n) = {\varphi }(n) n^{-1}$, with ${\varphi }(\cdot )$ slowly varying, so that ${\mathbf E}(\tau _1^{\epsilon })=\infty $ for all ${\epsilon }>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of ${\mathbf P}(\tau _k=n)$ when $n$ is at least the typical length of $\tau _k$. A recent renewal theorem in [22] is an immediate consequence: ${\mathbf P}(n\in \tau ) \sim{\mathbf P} (\tau _1=n)/{\mathbf P}(\tau _1 > n)^2$ as $n\to \infty $. If instead we only assume regular variation of ${\mathbf P}(n\in \tau )$ and slow variation of $U_n:= \sum _{k=0}^n {\mathbf P}(k\in \tau )$, we obtain a similar equivalence but with ${\mathbf P}(\tau _1=n)$ replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of ${\mathbf P}(\tau _k \leq n)$ when $n$ is much smaller than the typical length of $\tau _k$.

Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 66, 18 pp.

Dates
Received: 31 March 2016
Accepted: 6 November 2016
First available in Project Euclid: 26 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1480129233

Digital Object Identifier
doi:10.1214/16-EJP13

Subjects
Primary: 60K05: Renewal theory 60G50: Sums of independent random variables; random walks 60F10: Large deviations

Keywords
local limit theorem local large deviation renewal theorem reverse renewal theorems slowly varying tail distribution i.i.d. sums

Rights
Creative Commons Attribution 4.0 International License.

Citation

Alexander, Kenneth S.; Berger, Quentin. Local limit theorems and renewal theory with no moments. Electron. J. Probab. 21 (2016), paper no. 66, 18 pp. doi:10.1214/16-EJP13. https://projecteuclid.org/euclid.ejp/1480129233


Export citation

References

  • [1] K. S. Alexander and Q. Berger, Local asymptotics for the first intersection of two independent renewals, preprint, arXiv:1603.05531 [math.PR]
  • [2] S. Asmussen, Applied Probability and Queues, Second Edition, Applications of Mathematics 51, Springer-Verlag, New York, 2003.
  • [3] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variations, Cambridge University Press, Cambridge, 1987.
  • [4] F. Caravenna, The strong renewal theorem, preprint, arXiv:1507.07502 [math.PR]
  • [5] J. Chover, P. Ney and S. Wainger, Functions of probability measures, J. Anal. Math., 25 pp. 255–302, 1973.
  • [6] Z. Chi, Strong renewal theorem with infinite mean beyond local large deviations, Ann. Appl. Probab. 25 (2015), pp. 1513–1539.
  • [7] Z. Chi, Integral criteria for Strong Renewal Theorems with infinite mean, preprint, arXiv:1312.6089v3 [math.PR]
  • [8] D. A. Darling, The influence of the maximum term in the addition of independent random variables, Trans. Amer. Math. Soc. 73, pp. 95–107, 1952.
  • [9] D. Denisov, A. B. Dieker and V. Shneer, Large deviations for random walks under subexponentiality: the big-jump domain, Ann. Probab., 38 5, pp. 1946–1991, 2008.
  • [10] R. A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Relat. Fields, 107, pp. 451–465, 1997.
  • [11] R. A. Doney, The strong renewal theorem with infinite mean via local large deviations, preprint, arXiv:1507.06790 [math.PR]
  • [12] R. A. Doney and D. A. Korshunov, Local asymptotics for the time of first return to the origin of transient random walk, Stat. Probab. Letters, 81 5, pp. 363–365, 2011.
  • [13] K. B. Erickson, Strong renewal theorems with infinite mean, Transaction of the American Mathematical Society, 151, 1970.
  • [14] W. Feller, An introduction to probability theory and its applications, Vol. 1, 2nd edition, Wiley series in probability and mathematical statistics, John Wiley & Sons. Inc., New York-London-Sydney, 1966.
  • [15] A. Garsia and J. Lamperti, A discrete renewal theorem with infinite mean, Comm. Math. Helv. 37, pp. 221–234, 1963.
  • [16] G. Giacomin, Random polymer models, Imperial College Press, 2007.
  • [17] B. V. Gnedenko, Sur la distribution limite du terme maximum d’une série aléatoire, Ann. Math. (2) 44, pp. 423–453, 1943.
  • [18] B. V. Gnedenko and A. N. Kolmogorov, Limit Theorems for Sums of Independent Random Variables, Addison-Wesley, Cambridge, 1954.
  • [19] N. C. Jain and W. E. Pruitt, The range of random walk, Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3 pp. 31–50, Univ. California Press, Berkeley, 1972.
  • [20] Y. Kasahara, A limit theorem for sums of i.i.d. random variables with slowly varying tail probability, J. Math. Kyoto Univ. 26, pp. 437–443, 1986.
  • [21] H. Kesten, Ratio Theorems for Random Walks II, J. Analyse Math. 11, pp. 323–379, 1963.
  • [22] S. V. Nagaev, The Renewal Theorem in the Absence of Power Moments, Theory Probab. Appl., 56 No. 1, pp. 166–175, 2012.
  • [23] S. V. Nagaev, Large deviations of sums of independent random variables, Ann. Probab. 7, 745–789, 1979.
  • [24] S. V. Nagaev and V. I. Wachtel, On sums of independent random variables without power moments, Sib. Math. J., 49, No. 6, pp. 1091–1010, 2008.
  • [25] H. Teicher, Rapidly growing random walks and an associated stopping time, Ann. Probab. 7, pp. 1078–1081, 1979.
  • [26] S. Watanabe, A limit theorem for sums of i.i.d. random variables with slowly varying tail probability, Multivariate Analysis, V, Proc. Fifth Internat. Sympos., Pittsburgh, PA, 1978, pp. 249–261, North-Holland, Amsterdam, 1980.
  • [27] J. A. Williamson, Random walks and Riesz kernels, Pacific J. Math. 25, pp. 393–415, 1968.