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Limit theorems in fluctuation theory

Published online by Cambridge University Press:  01 July 2016

N. H. Bingham
N. H. Bingham*
Affiliation:
Westfield College, London
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Abstract

We shall be concerned here with limit theorems arising in the fluctuation theory of random walks, processes with stationary independent increments, recurrent events and regenerative phenomena. In Section 1 on discrete time, we consider limit theorems for ladder-points (Theorem 1) and for maxima of partial sums of random variables (Theorem 2), and discuss some related questions. In Section 2 (Theorems 3 to 6) we consider the analogues of these results in continuous time.

Keywords

SPITZER'S IDENTITYREGULAR VARIATIONTAUBERIAN THEOREMSLADDER POINTSRECURRENT EVENTSMAXIMA OF SUMS OF RANDOM VARIABLESp-FUNCTIONSSUPREMUM FUNCTIONALFIRST-PASSAGE FUNCTIONALREGENERATIVE PHENOMENA
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 3 , December 1973 , pp. 554 - 569
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Baxter, G. and Donsker, M. D. (1957) On the distribution of the supremum functional for processes with stationary independent increments. Trans. Amer. Math. Soc. 85, 73–87.Google Scholar
[2] Bingham, N. H. (1971) Limit theorems for occupation-times of Markov processes. Z. Wahrscheinlichkeitsth. 17, 1–22.Google Scholar
[3] Bingham, N. H. (1972) Limit theorems for regenerative phenomena, recurrent events and renewal theory. Z. Wahrscheinlichkeitsth. 21, 20–44.CrossRefGoogle Scholar
[4] Bingham, N. H. (1973) A class of limit theorems for Markov processes. Stochastic Geometry and Stochastic Analysis. Wiley, New York.Google Scholar
[5] Bingham, N. H. (1973) Maxima of partial sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitsth. To appear.Google Scholar
[6] Emery, D. J. (1972) Limiting behaviour of the distributions of the maxima of partial sums of certain random walks. J. Appl. Prob. 9, 572–579.CrossRefGoogle Scholar
[7] Feller, W. (1949) Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119.CrossRefGoogle Scholar
[8] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[9] Garsia, A. and Lamperti, J. (1963) A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37, 221–234.Google Scholar
[10] Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Theorems for Sums of Independent Random Variables. Addison-Wesley, London.Google Scholar
[11] Heyde, C. C. (1969) On extremal factorisation and recurrent events. J. R. Statist. Soc. B 31, 72–79.Google Scholar
[12] Heyde, C. C. (1969) On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Prob. 6, 419–429.Google Scholar
[13] Kingman, J. F. C. (1964) The stochastic theory of regenerative events. Z. Wahrscheinlichkeitsth. 2, 180–224.CrossRefGoogle Scholar
[14] Kingman, J. F. C. (1972) Regenerative Phenomena. Wiley, London.Google Scholar
[15] Lamperti, J. (1958) Limit theorems for stochastic processes. J. Math. Mech. 7, 433–450.Google Scholar
[16] Lamperti, J. (1958) An occupation-time theorem for stochastic processes. Trans. Amer. Math. Soc. 88, 380–387.Google Scholar
[17] Lamperti, J. (1962) An invariance principle in renewal theory. Ann. Math. Statist. 33, 685–696.Google Scholar
[18] Port, S. C. (1963) An elementary approach to fluctuation theory. J. Math. Anal. Appl. 6, 109–151.CrossRefGoogle Scholar
[19] Port, S. C. (1964) Some theorems on functionals of Markov chains. Ann. Math. Statist. 35, 1275–1290.Google Scholar
[20] Rogozin, B. A. (1966) On the distribution of functionals related to boundary value problems for processes with independent increments. Theor. Probability Appl. 11, 580–591.Google Scholar
[21] Rosén, B. (1962) On the asymptotic distribution of sums of independent identically distributed random variables. Ark. Mat. 4, 323–333.Google Scholar
[22] Rubinovitch, M. (1971) Ladder phenomena in stochastic processes with stationary independent increments. Z. Wahrscheinlichkeitsth. 20, 58–74.CrossRefGoogle Scholar
[23] Sparre-Andersen, E. (1954) On the fluctuation of sums of random variables, II. Math. Scand. 2, 195–223.Google Scholar
[24] Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–338.Google Scholar
[25] Spitzer, F. (1964) Principles of Random Walk, Van Nostrand, Princeton, N.J. CrossRefGoogle Scholar
[26] Zolotarev, V. M. (1957) Mellin-Stieltjes transform in probability theory. Theor. Probability Appl. 2, 433–460.Google Scholar
[27] Heyde, C. C. (1967) Some local limit results in fluctuation theory. J. Austral. Math. Soc. 7, 455–464.Google Scholar
[28] Heyde, C. C. (1968) A further generalisation of the arc-sine law. J. Austral. Math. Soc. 8, 369–372.Google Scholar
[29] Heyde, C. C. (1969) On a fluctuation theorem for processes with independent increments. Ann. Math. Statist. 40, 688–691.CrossRefGoogle Scholar
[30] Rogozin, B. A. (1971) The distributions of first ladder moment and height and fluctuation of a random walk. Theor. Probability Appl. 16, 575–595.Google Scholar
[31] Rogozin, B. A. (1972) The distribution of the first hit for stable and asymptotically stable walks on an interval. Theor. Probability Appl. 17, 332–339.CrossRefGoogle Scholar