Strong approximation of renewal processes. (English) Zbl 0545.60081
The author studies the first passage times of the partial sums of i.i.d. random variables and proves strong approximation results. As an application a Strassen type LIL is obtained. He also shows that a Bahadur-Kiefer type representation holds for the first passage time process.
Reviewer: A.Gut
References:
| [1] | Bahadur, R. R., A note on quantiles in large samples, Ann. Math. Statist., 37, 577-580 (1966) · Zbl 0147.18805 |
| [2] | Basu, A. K., Invariance theorems for first passage time random variables, Canad. Math. Bull., 15, 171-176 (1972) · Zbl 0245.60022 |
| [3] | Berkes, I.; Philipp, W., Approximation theorems for independent and weakly dependent random vectors, Ann. Probab., 7, 29-54 (1979) · Zbl 0392.60024 |
| [4] | Billingsley, P., Convergence of Probability Measures (1981), Wiley: Wiley New York · Zbl 0172.21201 |
| [5] | Csörgö, M.; Révész, P., Strong Approximation in Probability and Statistics (1981), Academic Press: Academic Press New York · Zbl 0539.60029 |
| [6] | Gut, A., A functional central limit theorem connected with extended renewal theory, Z. Wahrsch. Verw. Gebiete, 27, 123-129 (1973) · Zbl 0253.60029 |
| [7] | Heyde, C. C., Asymptomic renewal results for a natural generalization of classical renewal theory, J. Roy. Statist. Soc. Ser. B., 29, 141-150 (1967) · Zbl 0166.14003 |
| [8] | Iglehart, D. L.; Whitt, W., The equivalence of functional central limit theorems for counting processes and associated partial sums, Ann. Math. Statist., 42, 1372-1378 (1971) · Zbl 0226.60028 |
| [9] | Jain, N. C.; Jogdeo, K.; Stout, W. F., Upper and lower functions for martingales and mixing processes, Ann. Probab., 3, 119-145 (1975) · Zbl 0301.60026 |
| [10] | Kiefer, J., Deviations between the sample quantile process and sample D.F., (Puri, M. L., Nonparametric Techniques in Statistics (1970), Cambridge University Press: Cambridge University Press London) |
| [11] | Komlós, J.; Tusnády, G.; Major, P., An approximation of partial sums of independent R.V.’s and the sample D.F.I., Z. Wahrsch. Verw. Gebiete, 32, 111-135 (1975) · Zbl 0308.60029 |
| [12] | Komlós, J.; Tusnády, G.; Major, P., An approximation of partial sums of independent R.V.’s and the sample D.F.I., Z. Wahrsch. Verw. Gebiete, 34, 33-58 (1976) · Zbl 0307.60045 |
| [13] | Lévy, P., Processus Stochastique et Mouvement Brownien (1948), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0034.22603 |
| [14] | Major, P., The approximation of partial sums of independent r.v.’s, Z. Wahrsch. Verw. Gebiete, 35, 213-220 (1976) · Zbl 0338.60031 |
| [15] | W. Philipp and W. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. No. 161.; W. Philipp and W. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. No. 161. · Zbl 0361.60007 |
| [16] | Révész, P., On the increments of Wiener and related processes, Ann. Probab., 10, 613-622 (1982) · Zbl 0493.60038 |
| [17] | Vervaat, W., Functional limit theorems for processes with positive drift and their inverses, Z. Wahrsch. Verw. Gebiete, 23, 245-253 (1972) · Zbl 0238.60018 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
