## The Annals of Applied Probability

### The maximum of a random walk reflected at a general barrier

Niels Richard Hansen

#### Abstract

We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier.

#### Article information

Source
Ann. Appl. Probab., Volume 16, Number 1 (2006), 15-29.

Dates
First available in Project Euclid: 6 March 2006

https://projecteuclid.org/euclid.aoap/1141654279

Digital Object Identifier
doi:10.1214/105051605000000610

Mathematical Reviews number (MathSciNet)
MR2209334

Zentralblatt MATH identifier
1098.60044

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F10: Large deviations

#### Citation

Hansen, Niels Richard. The maximum of a random walk reflected at a general barrier. Ann. Appl. Probab. 16 (2006), no. 1, 15--29. doi:10.1214/105051605000000610. https://projecteuclid.org/euclid.aoap/1141654279

#### References

• Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
• Lee, C. R. and Ambros, V. (2001). An extensive class of small RNAs in Caenorhabditis elegans. Science 294 862–864.
• Feller, W. (1966). An Introduction to Probability Theory and Its Applications. II. Wiley, New York.
• Grossmann, S. and Yakir, B. (2004). Large deviations for global maxima of independent superadditive processes with negative drift and an application to optimal sequence alignments. Bernoulli 10 829–845.
• Hansen, N. R. (2005). Local maximal stack scores with general loop penalty function. Working paper. Available at www.math.ku.dk/~richard.
• Hansen, N. R. (2006). Local alignment of Markov chains. Ann. Appl. Probab. To appear. Available at www.math.ku.dk/~richard.
• Iglehart, D. L. (1972). Extreme values in the $\mathit{GI}/G/1$ queue. Ann. Math. Statist. 43 627–635.
• Karlin, S. and Dembo, A. (1992). Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. in Appl. Probab. 24 113–140.
• Mott, R. and Tribe, R. (1999). Approximate statistics of gapped alignments. J. Comput. Biol. 6 91–112.
• Neveu, J. (1965). Mathematical Foundations of the Calculus of Probability. Translated by Amiel Feinstein. Holden–Day, San Francisco.
• Siegmund, D. (1985). Sequential Analysis. Springer, New York.
• Siegmund, D. and Yakir, B. (2000). Approximate $p$-values for local sequence alignments. Ann. Statist. 28 3 657–680.
• Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia.