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The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

Published online by Cambridge University Press:  14 July 2016

N. Veraverbeke  and
J. L. Teugels
N. Veraverbeke
Affiliation:
Limburgs Universitair Centrum
J. L. Teugels
Affiliation:
Limburgs Universitair Centrum
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Abstract

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

Keywords

COMPLETE EXPONENTIAL CONVERGENCEMAXIMUM (SUPREMUM) OF A RANDOM WALKEXPONENTIAL RATE OF CONVERGENCEEXPANSIONS FOR THE CENTRAL LIMIT THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 4 , December 1976 , pp. 733 - 740
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Bahadur, R. R. and Rao, R. R. (1960) On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.Google Scholar
[2] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[3] Gnedenko, B. V. and Kolmogorov, A. N. (1968) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, Mass.Google Scholar
[4] Heathcote, C. R. (1967) Complete exponential convergence and some related topics. J. Appl. Prob. 4, 217–256.Google Scholar
[5] Iglehart, D. L. (1974) Random walks with negative drift conditioned to stay positive. J. Appl. Prob. 11, 742–751.Google Scholar
[6] Miller, H. D. (1966) Geometric ergodicity in a class of denumerable Markov chains. Z. Wahrscheinlichkeitsth. 4, 354–373.Google Scholar
[7] Petrov, V. V. (1965) On the probabilities of large deviations for sums of independent random variables. Theor. Prob. Appl. 10, 287–298.Google Scholar
[8] Prabhu, N. U. (1965) Stochastic Processes. Macmillan, New York.Google Scholar
[9] Veraverbeke, N. and Teugels, J. L. (1975) The exponential rate of convergence of the distribution of the maximum of a random walk. J. Appl. Prob. 12, 279–288.Google Scholar