Abstract
In this chapter, we study a random walk whose increments have a (right) heavy-tailed distribution with a negative mean. The maximum of such a random walk is almost surely finite, and our interest is in the tail asymptotics of the distribution of this maximum, for both infinite and finite time horizons; we are further interested in the local asymptotics for the maximum in the case of an infinite time horizon. We use direct probabilistic techniques and show that, under the appropriate subexponentiality conditions, the main reason for the maximum to be far away from zero is again that a single increment of the walk is similarly large.
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© 2011 Springer Science+Business Media, LLC
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Foss, S., Korshunov, D., Zachary, S. (2011). Maximum of Random Walk. In: An Introduction to Heavy-Tailed and Subexponential Distributions. Springer Series in Operations Research and Financial Engineering, vol 38. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9473-8_5
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DOI: https://doi.org/10.1007/978-1-4419-9473-8_5
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Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9472-1
Online ISBN: 978-1-4419-9473-8
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