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First-passage time asymptotics over moving boundaries for random walk bridges. (English) Zbl 1401.60082
Summary: We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent \(-\frac 12\), where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.
MSC:
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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[1] Aurzada, F.; Kramm, T., The first passage time problem over a moving boundary for asymptotically stable Lévy processes, J. Theoret. Prob., 29, 737-760, (2016) · Zbl 1354.60049
[2] Bolthausen, E., On a functional central limit theorem for random walks conditioned to stay positive, Ann. Prob., 4, 480-485, (1976) · Zbl 0336.60024
[3] Caravenna, F.; Chaumont, L., An invariance principle for random walk bridges conditioned to stay positive, Electron. J. Prob., 18, (2013) · Zbl 1291.60090
[4] Denisov, D.; Shneer, V., Asymptotics for the first-passage times of Lévy processes and random walks, J. Appl. Prob., 50, 64-84, (2013) · Zbl 1264.60031
[5] Denisov, D.; Sakhanenko, A.; Wachtel, V., First passage times for random walks with non-identically distributed increments, Ann. Appl. Prob, (2018) · Zbl 06975488
[6] Dobson, I.; Carreras, B. A.; Newman, D. E., Proceedings of the 37th Hawaii International Conference on System Sciences, A branching process approximation to cascading load-dependent system failure, (2004), IEEE
[7] Doney, R. A., Conditional limit theorems for asymptotically stable random walks, Z. Wahrscheinlichkeitsth., 70, 351-360, (1985) · Zbl 0573.60063
[8] Doney, R. A., Local behaviour of first passage probabilities, Prob. Theory Relat. Fields, 152, 559-588, (2012) · Zbl 1237.60036
[9] Donsker, M. D., An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc., 6, (1951) · Zbl 0042.37602
[10] Durrett, R. T.; Iglehart, D. L.; Miller, D. R., Weak convergence to Brownian meander and Brownian excursion, Ann. Prob., 5, 117-129, (1977) · Zbl 0356.60034
[11] Greenwood, P. E.; Novikov, A. A., One-sided boundary crossing for processes with independent increments, Theory Prob. Appl., 31, 221-232, (1987) · Zbl 0658.60103
[12] Greenwood, P.; Perkins, E., Limit theorems for excursions from a moving boundary, Theory Prob. Appl., 29, 731-743, (1985)
[13] Iglehart, D. L., Functional central limit theorems for random walks conditioned to stay positive, Ann. Prob., 2, 608-619, (1974) · Zbl 0299.60053
[14] Liggett, T. M., An invariance principle for conditioned sums of independent random variables, J. Math. Mech., 18, 559-570, (1968) · Zbl 0181.20502
[15] Novikov, A. A., The crossing time of a one-sided nonlinear boundary by sums of independent random variables, Theory Prob. Appl., 27, 688-702, (1982) · Zbl 0521.60055
[16] Novikov, A. A., The crossing time of a one-sided nonlinear boundary by sums of independent random variables, Theory Prob. Appl., 27, 688-702, (1983) · Zbl 0521.60055
[17] Petrov, V. V., Sums of Independent Random Variables, (1975), Springer: Springer, New York · Zbl 0322.60042
[18] Skorokhod, A. V., Limit theorems for stochastic processes with independent increments, Theory Prob. Appl., 2, 138-171, (1957) · Zbl 0097.13001
[19] Sloothaak, F.; Borst, S. C.; Zwart, A. P., Robustness of power-law behavior in cascading failure models, Stoch. Models., 34, 45-72, (2018) · Zbl 1386.90038
[20] Wachtel, V. I.; Denisov, D. E., An exact asymptotics for the moment of crossing a curved boundary by an asymptotically stable random walk, Theory Prob. Appl., 60, 481-500, (2016) · Zbl 1375.60088
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