On lerch's transcendent and the Gaussian random walk

Annals of Applied Probability

Volumn 17, Issue 2, 2007, Pages 421-439

  Janssen, A J E M ^a   Van Leeuwaarden, J S H ^b  

^a PHILIPS RESEARCH LABORATORIES   (Netherlands)

^b EURANDOM   (Netherlands)

Author keywords

All time maximum; Euler Maclaurin summation; Gaussian random walk; Lerch's transcendent; Riemann zeta function; Spitzer's identity

Indexed keywords

EID: 35348990154     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051606000000781     Document Type: Article

References (27)

1. ABRAMOWITZ, M. and STEGUN, I. A. (1970). Handbook of Mathematical Functions, 9th printing. Dover, New York.
2. ASMUSSEN, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York. MR1978607
3. ASMUSSEN, S., GLYNN, P. and PITMAN, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Probab. 5 875-896. MR1384357
4. BLANCHET, J. and GLYNN, P. (2006). Complete corrected diffusion approximations for the maximum of a random walk. Ann. Appl. Probab. 16 951-983. MR2244438
5. BOERSMA, J. and DEMPSEY, J. P. (1992). On the numerical evaluation of Legendre's chifunction. Math. Comput. 59 157-163. MR1134715
6. BROADIE, M., GLASSERMAN, P. and KOU, S. (1997). A continuity correction for discrete barrier options. Math. Finance 7 325-349. MR1482707
7. DE BRUIJN, N. G. (1981). Asymptotic Methods in Analysis. Dover, New York. MR0671583
8. CALVIN, J. (1995). Average performance of nonadaptive algorithms for global optimization. J. Math. Anal. Appl. 191 608-617. MR1325821
9. CHANG, J. T. (1992). On moments of the first ladder height of random walks with small drift. Ann. Appl. Probab. 2 714-738. MR1177906
10. CHANG, J. T. and PERES, Y. (1997). Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Probab. 25 787-802. MR1434126
11. CHERNOFF, H. (1965). Sequential test for the mean of a normal distribution. IV. (Discrete case). Ann. Math. Statist. 36 55-68. MR0170443
12. COMTET, A. and MAJUMDAR, S. N. (2005). Precise asymptotics for a random walker's maximum. J. Statist. Mech. Theory Experiment P06013.
13. ERDÉLYI, A., MAGNUS, W., OBERHETTINGER, F. and TRICOMI, F. G. (1953). HigherTranscendental Functions. I. McGraw-Hill, New York.
14. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications, II, 2nd ed. Wiley, New York.
15. GLASSER, M. L. and BOERSMA, J. (1981). Problem 602. Nieuw Archiefvoor Wiskunde 29 301-304.
16. HALFIN, S. and WHITT, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567-588. MR0629195
17. HARDY, G. H. (1905). A method for determining the behavior of certain classes of power series near a singular point on the circle of convergence. Proc. London Math. Soc. 2 381-389.
18. JELENKOVIĆ, P., MANDELBAUM, A. and MOMČILOVIĆ, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Systems 47 54-69. MR2074672
19. KINGMAN, J. F. C. (1962). On queues in heavy traffic. J. Roy. Statist. Soe. Ser. B 24 383-392. MR0148146
20. KINGMAN, J. F. C. (1965). The heavy traffic approximation in the theory of queues. In Proc. Symp. on Congestion Theory (W. L. Smith and W. Wilkinson, eds.) 137-169. Univ. North Carolina Press, Chapel Hill. MR0198566
21. LAI, T. L. (1976). Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab. 4 51-66. MR0391265
22. SIEGMUND, D. (1979). Sequential analysis: Tests and confidence intervals. Adv. in Appl. Probab. 11 701-719. MR0799155
23. SIEGMUND, D. (1985). Corrected Diffusion Approximations in Certain Random Walk Problems. Springer, New York.
24. SPITZER, F. L. (1956). A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 323-339. MR0079851
25. SPITZER, F. L. (1960). A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94 150-169. MR0111066
26. WHITT, W. (1974). Heavy traffic limit theorems for queues: A survey. Lecture Notes in Econom. and Math. Systems 98 307-350. Springer, Berlin. MR0394935
27. WHITTAKER, E. T. and WATSON, G. N. (1963). A Course of Modern Analysis, 4th ed. Cambridge Univ. Press. MR1424469