## Electronic Journal of Probability

### Local large deviations and the strong renewal theorem

#### Abstract

We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha$. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long-standing problem, which dates back to the 1962 paper of Garsia and Lamperti [GL62] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Wil68] for general random walks.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 72, 48 pp.

Dates
Accepted: 12 May 2019
First available in Project Euclid: 28 June 2019

https://projecteuclid.org/euclid.ejp/1561687605

Digital Object Identifier
doi:10.1214/19-EJP319

Mathematical Reviews number (MathSciNet)
MR3978222

Zentralblatt MATH identifier
07089010

#### Citation

Caravenna, Francesco; Doney, Ron. Local large deviations and the strong renewal theorem. Electron. J. Probab. 24 (2019), paper no. 72, 48 pp. doi:10.1214/19-EJP319. https://projecteuclid.org/euclid.ejp/1561687605

#### References

• [AA87] K. K. Anderson and K. B. Athreya, A renewal theorem in the infinite mean case, Ann. Probab. 15 (1987), 388–393.
• [Ber17] Q. Berger, Notes on Random Walks in the Cauchy Domain of Attraction, Probab. Theory Relat. Fields (to appear), preprint (2017), arXiv:1706.07924 [math.PR].
• [Ber19] Q. Berger, Strong renewal theorems and local large deviations for multivariate random walks and renewals, Electron. J. Probab. 24 (2019), no. 46, 47 pp.
• [Ber96] J. Bertoin, Lévy Processes, Cambridge University Press (1996).
• [BD97] J. Bertoin and R. A. Doney, Spitzer’s condition for random walks and Levy processes, Ann. Inst. H. Poincaré 32 (1997), 167–178.
• [BGT89] N. H. Bingham, C. H. Goldie and J. L. Teugels, Regular variation, Cambridge University Press (1989).
• [Car15] F. Caravenna, The strong renewal theorem, preprint (2015), arXiv:1507.07502 [math.PR].
• [CSZ16] F. Caravenna, R. Sun and N. Zygouras, The continuum disordered pinning model, Probab. Theory Related Fields 164 (2016), 17–59.
• [Chi15] Z. Chi, Strong renewal theorem with infinite mean beyond local large deviations, Ann. Appl. Probab. 25 (2015), 1513–1539.
• [Chi13] Z. Chi, Integral criteria for Strong Renewal Theorems with infinite mean, preprint, arXiv:1312.6089v3 [math.PR].
• [Chi18] Z. Chi, On a Multivariate Strong Renewal Theorem, J. Theor. Probab. 31 (2018), 1235–1272.
• [DSW18] D. Denisov, A. Sakhanenko, V. Wachtel, First-passage times over moving boundaries for asymptotically stable walks, Theory Probab. Appl. 63 (2019), 613–633.
• [DN17] D. Dolgopyat, P. Nándori, Infinite measure renewal theorem and related results, Bulletin LMS 51 (2019), 145–167.
• [DN18] D. Dolgopyat, P. Nándori, Infinite measure mixing for some mechanical systems, preprint (2018), arXiv:1812.01174 [math.DS].
• [Don97] R. A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Rel. Fields 107 (1997), 451–465.
• [Don15] R. A. Doney, The strong renewal theorem with infinite mean via local large deviations, preprint (2015), arXiv:1507.06790 [math.PR].
• [DW18] J. Duraj, V. Wachtel, Green function of a random walk in a cone, preprint (2018), arXiv:1807.07360 [math.PR].
• [Eri70] K. B. Erickson, Strong renewal theorems with infinite mean, Trans. Amer. Math. Soc. 151 (1970), 263–291.
• [Eri71] K. B. Erickson, A renewal theorem for distributions on $R^{1}$ without expectation, Bull. Amer. Math. Soc. 77 (1971), 406–410.
• [FMMV19] L. R. G. Fontes, D. H. U. Marchetti, T. S. Mountford, M. E. Vares, Contact process under renewals I, Stochastic Process. Appl. 129 (2019), 2903–2911.
• [GL62] A. Garsia and J. Lamperti, A discrete renewal theorem with infinite mean, Comm. Math. Helv. 37, 221–234, 1962.
• [Gia07] G. Giacomin, Random polymer models, Imperial College Press, World Scientific (2007).
• [Gia11] G. Giacomin, Disorder and Critical Phenomena Through Basic Probability Models, École d’Été de Probabilités de Saint-Flour XL–2010, Lecture Notes in Mathematics 2025, Springer, 2011.
• [Hol09] F. den Hollander, Random Polymers, École d’Été de Probabilités de Saint-Flour XXXVII–2007, Springer (2009).
• [Kev17] P. Kevei, Implicit renewal theory in the arithmetic case, J. Appl. Probab. 54 (2017), 732–749.
• [Kol17] B. Kołodziejek, The left tail of renewal measure, Statist. Probab. Letters 129 (2017), 306–310.
• [MT17] I. Melbourne and D. Terhesiu, Renewal theorems and mixing for non Markov flows with infinite measure, preprint (2017), Ann. Inst. H. Poincaré (to appear), arXiv:1701.08440 [math.DS].
• [Nag79] A. V. Nagaev, Large deviations of sums of independent random variables, Ann. Probab. 7 (1979), 745–789.
• [Uch18] K. Uchiyama, On the ladder heights of random walks attracted to stable laws of exponent 1, Electron. Commun. Probab. 23 (2018), paper no. 23, 12 pp.
• [VT13] V. A. Vatutin and V. Topchii, A key renewal theorem for heavy tail distributions with $\beta \in (0,0.5]$, Theory Probab. Appl. 58 (2013), 387–396.
• [Wil68] J. A. Williamson, Random walks and Riesz kernels, Pacific J. Math. 25 (1968), 393–415.