K. S. Alexander and Q. Berger, Local asymptotics for the first intersection of two independent renewals, preprint

. Asmussen, Applied Probability and Queues, Second Edition, Applications of Mathematics, vol.51, 2003.

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variations, 1987.
DOI : 10.1017/CBO9780511721434

F. Caravenna, The strong renewal theorem, preprint, arXiv:1507, p.7502

J. Chover, P. Ney, and S. Wainger, Functions of probability measures, Journal d'Analyse Math??matique, vol.26, issue.1, pp.255-302, 1973.
DOI : 10.1007/BF02790433

Z. Chi, Strong renewal theorems with infinite mean beyond local large deviations, The Annals of Applied Probability, vol.25, issue.3, pp.1513-1539, 2015.
DOI : 10.1214/14-AAP1029

Z. Chi, Integral criteria for Strong Renewal Theorems with infinite mean, preprint, arXiv:1312.6089v3 [math

D. A. Darling, The influence of the maximum term in the addition of independent random variables, Transactions of the American Mathematical Society, vol.73, issue.1, pp.95-107, 1952.
DOI : 10.1090/S0002-9947-1952-0048726-0

D. Denisov, A. B. Dieker, and V. Shneer, Large deviations for random walks under subexponentiality: The big-jump domain, The Annals of Probability, vol.36, issue.5, pp.1946-1991, 2008.
DOI : 10.1214/07-AOP382

R. A. Doney, One-sided local large deviation and renewal theorems in the case of infinite mean, Probab. Theory Relat, pp.451-465, 1997.

R. A. Doney, The strong renewal theorem with infinite mean via local large deviations, preprint

R. A. Doney and D. A. Korshunov, Local asymptotics for the time of first return to the origin of transient random walk, Statistics & Probability Letters, vol.81, issue.9, pp.363-365, 2011.
DOI : 10.1016/j.spl.2011.04.017

K. B. Erickson, Strong renewal theorems with infinite mean, Transaction of the, 1970.

W. Feller, An introduction to probability theory and its applications Wiley series in probability and mathematical statistics, 1966.

A. Garsia and J. Lamperti, A discrete renewal theorem with infinite mean, Commentarii Mathematici Helvetici, vol.37, issue.1, pp.221-234, 1963.
DOI : 10.1007/BF02566974

G. Giacomin, Random polymer models, 2007.
DOI : 10.1142/p504

URL : https://hal.archives-ouvertes.fr/hal-00155080

B. V. Gnedenko, Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire, The Annals of Mathematics, vol.44, issue.3, pp.423-453, 1943.
DOI : 10.2307/1968974

B. V. Gnedenko and A. N. Kolmogorov, Limit Theorems for Sums of Independent Random Variables, 1954.

N. C. Jain and W. E. Pruitt, On the range of random walk, Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3 pp, pp.31-50, 1972.
DOI : 10.1007/BF02771217

Y. Kasahara, A limit theorem for sums of i.i.d. random variables with slowly varying tail probability, Journal of Mathematics of Kyoto University, vol.26, issue.3, pp.437-443, 1986.
DOI : 10.1215/kjm/1250520877

H. Kesten, Ratio theorems for random walks II, Journal d'Analyse Math??matique, vol.17, issue.1, pp.323-379, 1963.
DOI : 10.1007/BF02789990

S. V. Nagaev, The Renewal Theorem in the Absence of Power Moments, Theory Probab, Appl, vol.56, issue.1, pp.166-175, 2012.

S. V. Nagaev, Large Deviations of Sums of Independent Random Variables, The Annals of Probability, vol.7, issue.5, pp.745-789, 1979.
DOI : 10.1214/aop/1176994938

S. V. Nagaev and V. I. , On sums of independent random variables without power moments, Math. J, vol.49, issue.6, pp.1091-1010, 2008.

H. Teicher, Rapidly Growing Random Walks and an Associated Stopping Time, The Annals of Probability, vol.7, issue.6, pp.1078-1081, 1979.
DOI : 10.1214/aop/1176994903

S. Watanabe, A limit theorem for sums of i.i.d. random variables with slowly varying tail probability, Multivariate Analysis, Proc. Fifth Internat. Sympos, pp.249-261, 1978.

J. A. Williamson, Random walks and Riesz kernels, Pacific Journal of Mathematics, vol.25, issue.2, pp.393-415, 1968.
DOI : 10.2140/pjm.1968.25.393