# zbMATH — the first resource for mathematics

Basic renewal theorems for random walks with widely dependent increments. (English) Zbl 1230.60095
Authors’ abstract: “We derive some basic renewal theorems for random walks with widely dependent increments, which contain some common negatively dependent random variables (r.v.s), some positively dependent r.v.s and some others. For this purpose, we investigate uniform integrability for related counting processes and the strong law of large numbers for widely dependent r.v.s.”

##### MSC:
 60K05 Renewal theory 60F15 Strong limit theorems 60F25 $$L^p$$-limit theorems
Full Text:
##### References:
 [1] Block, H.W.; Savits, T.H.; Shaked, M., Some concept of negative dependence, Ann. probab., 10, 765-772, (1982) · Zbl 0501.62037 [2] Chen, Y.; Chen, A.; Ng, K.W., The strong law of large numbers for extend negatively dependent random variables, J. appl. probab., 47, 908-922, (2010) · Zbl 1213.60058 [3] Chen, Y.; Yuen, K.C.; Ng, K.W., Precise large deviations of random sums in presence of negative dependence and consistent variation, Methodol. comput. appl. probab., (2010) [4] Chow, Y.S.; Lai, T.L., Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings, Trans. amer. math. soc., 208, 51-72, (1975) · Zbl 0335.60021 [5] Doob, J.L., Renewal theory from the point of view of the theory of probability, Trans. amer. math. soc., 63, 422-438, (1948) · Zbl 0041.45405 [6] Ebrahimi, N.; Ghosh, M., Multivariate negative dependence, Comm. statist. theory methods, 10, 307-337, (1981) · Zbl 0506.62034 [7] Gut, A., On the moments and limit distributions of some first passage times, Ann. probab., 2, 277-308, (1974) · Zbl 0278.60031 [8] Gut, A., Stopped random walks. limit theorems and applications. applied probability, Appl. probab. trust, vol. 5, (1988), Springer-Verlag New York · Zbl 0634.60061 [9] Heyde, C.C., Some renewal theorems with application to a first passage problem, Ann. math. statist., 37, 699-710, (1966) · Zbl 0143.19102 [10] Kesten, H.; Maller, R.A., Two renewal theorems for general random walks tending to infinity, Probab. theory related fields, 106, 1-38, (1996) · Zbl 0855.60080 [11] Lai, T.L., On uniform integrability in renewal theory, Bull. inst. math. acad. sin., 3, 1, 99-105, (1975) · Zbl 0329.60056 [12] Liu, L., Precise large deviations for dependent random variables with heavy tails, Statist. probab. lett., 79, 9, 1290-1298, (2009) · Zbl 1163.60012 [13] Rolski, T.; Schmidili, H.; Schmidt, V., Stochastic processes for insurance and finance, (1999), Wiley New York [14] Wang, K.; Wang, Y.; Gao, Q., Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. comput. appl. probab., (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.