Abstract
We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,∞) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided the fourth moment is finite.
References
Addario-Berry, L., Reed, B.: Ballot theorems for random walks with finite variance. Preprint 28 February 2008. ArXiv math.PR/0802.2491
Ahsanullah M., Nevzorov V.B.: Ordered Random Variables. Nova Science, Huntington (2001)
Arak, T.V.: The distribution of the maximum of the successive sums of independent random variables (Russian). Teor. Verojatnost. i Primenen. 19, 257–277 (1974) [English summary. English translation in Theor. Probab. Appl. 19, 245–266 (1974)]
Csörgő M., Révész P.: Strong Approximations in Probability and Statistics. Academic Press, New York (1981)
Erdős P., Kac M.: On certain limit theorems of the theory of probability. Bull. Am. Math. Soc. 52, 292–302 (1946)
Ford, K.: Du théorème de Kolmogorov sur les distributions empiriques à la théorie des nombres. L’héritage de Kolmogorov en mathématiques (French), pp. 111–120. Editions Belin, Paris (2004)
Ford, K.: From Kolmogorov’s Theorem on Empirical Distribution to Number Theory, Kolmogorov’s Legacy in Mathematics (English), pp. 97–108. Editions Belin, Springer, Paris (2007)
Ford K.: The distribution of integers with a divisor in a given interval. Ann. Math. 168, 367–433 (2008)
Ford K.: Sharp probability estimates for generalized Smirnov statistics. Monatsh. Math. 153, 205–216 (2008)
Gnedenko, B.V., Kolmogorov, A.N. (1968) Limit distributions for sums of independent random variables. Translated from the original 1949 Russian edition, annotated, and revised by K.L. Chung. With appendices by J.L. Doob and P.L. Hsu. Revised edition. Addison–Wesley, Reading (1968)
Kolmogorov A.N.: Sulla determinazione empirica di una legge di distribuzione (on the empirical determination of a distribution law). Giorn. Ist. Ital. Attuar. 4, 83–91 (1933)
Kozlov, M.: On the asymptotic probability of nonextinction for a critical branching process in a random environment. Teor. Verojatnost. i Primen. 21, 813–825 (1976) [Russian. English summary. English translation in Theor. Probab. Appl. 21, 791–804 (1976)]
Nagaev, S.V.: The rate of convergence of the distribution of the maximum of sums of independent random variables (Russian). Teor. Verojatnost. i Primenen 15, 320–326 (1970) [English Translation in Theor. Probability Appl. 15, 309–314 (1970)]
Pemantle R., Peres Y.: Critical random walk in random environment on trees. Ann. Probab. 23(1), 105–140 (1995)
Philipp W.: Invariance principles for independent and weakly dependent random variables. Dependence in probability and statistics (Oberwolfach, 1985). Prog. Probab. Statist 11, 225–268 (1986)
Rényi A: On the theory of order statistics. Acta Math. Acad. Sci. Hung. 4, 191–232 (1953)
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The author was supported by NSF grants DMS-0301083 and DMS-0555367.
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Ford, K. Sharp probability estimates for random walks with barriers. Probab. Theory Relat. Fields 145, 269–283 (2009). https://doi.org/10.1007/s00440-008-0168-4
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DOI: https://doi.org/10.1007/s00440-008-0168-4
Keywords
- Random walk
- Barrier
- Ballot theorems
Mathematics Subject Classification (2000)
- Primary 60G50