Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T16:50:08.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems in fluctuation theory

Published online by Cambridge University Press:  01 July 2016

N. H. Bingham
N. H. Bingham*
Affiliation:
Westfield College, London
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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