A limit theorem for sums of random number of i. i. d. random variables and its application to occupation times of Markov chains

Yuji KASAHARA

doi:10.2969/jmsj/03720197

Yuji KASAHARA

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JOURNAL FREE ACCESS

1985 Volume 37 Issue 2 Pages 197-205

DOI https://doi.org/10.2969/jmsj/03720197

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Published: 1985 Received: March 02, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : A limit theorem for sums of random number of i.i.d. random variables and its application to occupation times of Markov chains
Right : A limit theorem for sums of random number of i. i. d. random variables and its application to occupation times of Markov chains

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) K. L. Chung, Contributions to the theory of Markov chains II, Trans. Amer. Math. Soc., 76 (1954), 397-419.
2) D. A. Darling, The influence of the maximum term in the addition of independent random variables, Trans. Amer. Math. Soc., 73 (1952), 95-107.
3) R. L. Dobrusin, Two limit theorems for the simplest random walk on a line, Uspehi Mat. Nauk, 10 no 3 (1955), 139-146.
4) W. Doeblin, Sur deux problèmes de M. Kolmogoroff concertant les chaines dénombrables, Bull. Soc. Math. France, 66 (1938), 210-220.
5) W. Feller, An introduction to probability theory and its applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
6) Y. Kasahara, A limit theorem for slowly increasing occupation times, Ann. Probability, 10 (1982), 728-736.
7) Y. Kasahara, Another limit theorem for slowly increasing occupation times, J. Math. Kyoto Univ., 24 (1984), 507-520.
8) Y. Kasahara, Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions, J. Math. Kyoto Univ., 521-538.
9) Y. Kasahara, A note on sums and maxima of independent, identically distributed random variables, Proc. Japan Acad. Ser. A, 60 (1984), 353-356.
10) H. Kesten, Occupation times for Markov and semi-Markov chains, Trans. Amer. Math. Soc., 103 (1962), 82-112.
11) T. Lindvall, Weak convergence of probability measures and random functions in the function space D0, ∞), J. Appl. Probability, 10 (1973), 109-121.
12) F. Spitzer, Principles of random walk, D. Van Nostrand, Princeton, 1964.
13) S. I. Resnick, Inverses of extremal processes, Advances in Appl. Probability, 6 (1974), 392-406.
14) S. Watanabe, A limit theorem for sums of non-negative i.i.d. random variables with slowly varying tail probabilities, Proc. 5th International Conf. Multivariate Anal., 1980, 249-261.

Right : [1] K. L. Chung, Contributions to the theory of Markov chains II, Trans. Amer. Math. Soc., 76 (1954), 397-419.
[2] D. A. Darling, The influence of the maximum term in the addition of independent random variables, Trans. Amer. Math. Soc., 73 (1952), 95-107.
[3] R. L. Dobrusin, Two limit theorems for the simplest random walk on a line, Uspehi Mat. Nauk, 10 no 3 (1955), 139-146.
[4] W. Doeblin, Sur deux problèmes de M. Kolmogoroff concertant les chaines dénombrables, Bull. Soc. Math. France, 66 (1938), 210-220.
[5] W. Feller, An introduction to probability theory and its applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
[6] Y. Kasahara, A limit theorem for slowly increasing occupation times, Ann. Probability, 10 (1982), 728-736.
[7] Y. Kasahara, Another limit theorem for slowly increasing occupation times, J. Math. Kyoto Univ., 24 (1984), 507-520.
[8] Y. Kasahara, Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions, J. Math. Kyoto Univ., 521-538.
[9] Y. Kasahara, A note on sums and maxima of independent, identically distributed random variables, Proc. Japan Acad. Ser. A, 60 (1984), 353-356.
[10] H. Kesten, Occupation times for Markov and semi-Markov chains, Trans. Amer. Math. Soc., 103 (1962), 82-112.
[11] T. Lindvall, Weak convergence of probability measures and random functions in the function space D[0,∞), J. Appl. Probability, 10 (1973), 109-121.
[12] F. Spitzer, Principles of random walk, D. Van Nostrand, Princeton, 1964.
[13] S. I. Resnick, Inverses of extremal processes, Advances in Appl. Probability, 6 (1974), 392-406.
[14] S. Watanabe, A limit theorem for sums of non-negative i. i. d. random variables with slowly varying tail probabilities, Proc. 5th International Conf. Multivariate Anal., 1980, 249-261.

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