Electronic Journal of Probability

Strong renewal theorems and local large deviations for multivariate random walks and renewals

Quentin Berger

Full-text: Open access


We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 46, 47 pp.

Received: 3 August 2018
Accepted: 17 April 2019
First available in Project Euclid: 10 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60K05: Renewal theory 60F15: Strong theorems 60F10: Large deviations

multivariate random walks strong renewal theorems local large deviations

Creative Commons Attribution 4.0 International License.


Berger, Quentin. Strong renewal theorems and local large deviations for multivariate random walks and renewals. Electron. J. Probab. 24 (2019), paper no. 46, 47 pp. doi:10.1214/19-EJP308. https://projecteuclid.org/euclid.ejp/1557453645

Export citation


  • [1] K. Alexander and Q. Berger, Pinning a renewal on a quenched renewal, Electron. J. Probab., Vol. 23, no 6, 48 pp., 2018.
  • [2] K. K. Anderson and K. B. Athreya, A note on conjugate $\Pi $-variation and a weak limit theorem for the number of renewals, Stat. Probab. Letters, Vol. 6, pp. 151–154, 1988.
  • [3] Q. Berger, Notes on Random Walks in the Cauchy domain of attraction, Probab. Theory Relat. Fields, to appear.
  • [4] Q. Berger, G. Giacomin and M. Khatib, DNA melting structures for the generalized Poland-Scheraga model, ALEA, Lat. Am. J. Probab. Math. Stat., Vol. 15, pp. 993–1025, 2018.
  • [5] Q. Berger, G. Giacomin and M. Khatib, Disorder and denaturation transition in the generalized Poland-Scheraga Model, preprint arXiv:1807.11397 [math.PR].
  • [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, second ed., Encyclopedia Math. Appl., Vol. 27, Cambridge University Press, Cambridge, 1989.
  • [7] F. Caravenna and R. A. Doney, Local large deviations and the strong renewal theorem, preprint, arXiv:1612.07635.
  • [8] H. Carlsson and S. Wainger, On the multi-dimensional renewal theorem, J. Math. Anal. Appl., Vol. 100, pp. 316–322, 1984.
  • [9] Z. Chi, On a multivariate strong renewal theorem, J. Theor. Probab., Vol. 31, Issue 3, pp. 1235–1272, 2017.
  • [10] D. Denisov, A. B. Dieker and V. Shneer, Large deviations for random walks under subexponentiality: the big-jump domain, Ann. Probab., Vol. 36, no 5, pp. 1946–1991, 2008.
  • [11] R. A. Doney, A bivariate Local Limit Theorem, Jour. Multivariate Anal., Vol. 36, no 1, pp. 95–102, 1991.
  • [12] R. A. Doney, An analogue of renewal theorems in higher dimensions, Proc. London Math. Soc., Vol. 16, no 3, pp. 669-684, 1966.
  • [13] R. A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Relat. Fields, Vol. 107, no 4, pp. 451–465, 1997.
  • [14] K. B. Erickson, Strong renewal theorems with infinite mean, Transaction of the Americ. Math. Soc., Vol. 151, no 1, pp. 263–291,1970.
  • [15] W. Feller, An Introduction to Probability Theory and its Applications, Vol II, Wiley, New-York, 2nd ed., 1979.
  • [16] T. Garel and H. Orland, Generalized Poland-Scheraga model for DNA hybridization, Biopolymers, Vol. 75, no 6, pp. 453-467, 2004.
  • [17] A. Garsia and J. Lamperti, A discrete renewal theorem with infinite mean, Comm. Math. Helv., Vol. 37, no 1, pp. 221–234, 1962.
  • [18] G. Giacomin, M. Khatib, Generalized Poland-Scheraga denaturation model and two-dimensional renewal processes, Stoch. Proc. Appl., Vol. 127, no 2, pp. 526–573, 2017.
  • [19] P. S. Griffin, Matrix normalized sums of independent identically distributed random vectors, Ann. Probab. Vol. 14, no 1, pp. 224–246, 1986.
  • [20] L. de Haan, E. Omey and S. I. Resnick, Domains of attraction and regular variation in ${{\mathbb R}}^d$, J. Multivariate Anal., Vol. 14, Issue 1, pp. 17–33, 1984.
  • [21] L. de Haan and S. I. Resnick, Conjugate $\Pi $-variation and process inversion, Ann. Probab., Vol. 7, pp. 1028–1035, 1979.
  • [22] M. Hahn and M. Klass, The generalized domain of attraction of spherically symmetric stable laws on ${{\mathbb R}}^d$, In Proceedings Conf. Probab. Theory on Vector Spaces II, Lecture notes in Math., Vol. 828, pp. 52-81, Springer-Verlag, New-York/Berlin, 1979.
  • [23] M. Hahn and M. Klass, Affine normability of partial sums of i.i.d. random vectors: a characterization, Z. Warsch. Verw. Gebiete, Vol. 69, no 4, pp. 479–506, 1985.
  • [24] W. N. Hudson, Operator-stable distributions and stable marginals, J. Multivar. Anal., Vol. 10, no 1, pp. 26–37, 1980.
  • [25] H. Hult, F. Lindskog, T. Mikosch and G. Samorodnitsky, Functional large deviations for multivariate regularly varying random walks, Ann. App. Probab., Vol. 15, no. 4, pp. 2651–2680, 2005.
  • [26] P. Lévy, Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1937.
  • [27] M. M. Meerschaert, Regular variation in ${{\mathbb R}}^k$, Proceedings Amer. Math. Soc., Vol. 102, no 2, p. 341–348, 1988.
  • [28] M. M. Meerschaert, Regular variation in ${{\mathbb R}}^k$ and vector-normed domains of attraction, Stat. Probab. Letters, Vol. 11, no 4, pp. 287–289, 1991.
  • [29] M. M. Meerschart and H.-S. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley, 2001.
  • [30] M. M. Meerschart and H.-S. Scheffler, One-dimensional marginals of operator stable laws and their domains of attraction, Publ. Math. Debrecen, Vol. 55, no 3-4, pp. 487-499, 1999.
  • [31] A. V. Nagaev, Large deviations of sums of independent random variables, Ann. Probab., Vol. 7, no 5, pp. 745–789, 1979.
  • [32] A. V. Nagaev, Renewal theorems in $\mathbb{R} ^d$, Theory Probab. App., Vol. 24, no 3, pp. 572-581, 1980.
  • [33] A. V. Nagaev and A. Zaigraev, New large-deviation local theorems for sums of independent and identically distributed random vectors when the limit distribution is $\alpha $-stable, Bernoulli, Vol. 11, no 4, pp. 665–687, 2005.
  • [34] P. Ney and F. Spitzer, The Martin boundary for random walk, Trans. Amer. Math. Soc., Vol. 121, no 1, pp. 116–132, 1966.
  • [35] S. Resnick and P. E. Greenwood, A bivariate stable characterization and domains of attraction, J. Multivariate Anal., Vol. 9, no 2, pp. 206–221, 1979.
  • [36] E. L. Rvaceva, On domains of attractions of multidimensional distributions, Selected Transl. Math. Stat. Probab. Theory, Vol. 2, pp. 183–205, 1962.
  • [37] M. Sharpe, Operator-Stable Probability Distributions on Vector Groups, Trans. Amer. Math. Soc., Vol. 136, pp. 51–65, 1969.
  • [38] F. Spitzer, Principles of random walks, 2nd edn (Springer, Berlin), 1976.
  • [39] A. Stam, Renewal theory in $r$ dimensions, Compositio Math., Vol. 21, no 4, pp. 383-399, 1969.
  • [40] K. Uchiyama, Green’s functions for random walks on ${\mathbb Z}^N$, Proc. Lond. Math. Soc., Vol. 77, no 3, pp. 215–240, 1998.
  • [41] J. A. Williamson, Random walks and Riesz kernels, Pacific J. Math., Vol. 25, no 2, pp. 393–415, 1968.
  • [42] A. Zaigraev, Multivariate large deviations with stable limit laws, Probab. Math. Stat., Vol. 19, no 2, pp. 323-335, 1999.