Abstract
Let S n be a random walk with independent increments. We state necessary and sufficient conditions under which the relation P(S n — median(S n) > ɛb n) = o(d n) holds for any (for some) ɛ > 0 in the following cases: b n = is the same as in the previous case, or . Based on these particular cases we describe a method, which indeed allows us to derive such necessary and sufficient conditions for more general classes of b n and d n.
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References
Amosova N.N. (1978) On a problem of the convergence rates in one-sided law of large numbers. Izv.Vysshich.Uchebn.Zaved.,N 10, 3–6 (In Russian).
Amosova N.N. (1979) On probabilities of moderate deviations. Zap.Nauch.Semin. LOMI, 85, 6–16.
Esseen C.-G. (1968) On the concentration function of a sum of independent random variables. Z. Wahrscheinlich, verw. Geb., 9, 290–308.
Heyde C.C., Rohatgi V.K. (1967) A pair of complementary theorems on convergence rates in the law of large numbers. Proc. Camb. Phil. Soc, 63, 73–82.
Martikainen A.I. (1982) One-sided variants of the law of large numbers, strong law and rates of convergence. Proc. 16th All-Union Sunnol on Probab. Theory and Math. Statist.,Bakuriani,Febr. 26-March 5, 1982: Metsniereba, Tbilisi, 45-61 (in Russian).
Martikainen A.I. (1992) On convergence rates in one-sided law of large numbers. Ann. Acad. Sei. Fenn., Ser. A. I. Mathem. 17, 81–84.
Martikainen A.I. (1993) General one-sided laws of the iterated logarithm for random walks. Preprint, Philipps-Universität Marburg, Reihe Mathematik, N 33.
Petrov V.V. Sums of independent random variables. Berlin-Heidelberg-New-York: Springer-Verlag.
Petrov V.V., Shirokova I.V. (1973) On exponential rate of convergence in the law of large numbers. Vestnik Leningrad Univ. Mat. Mekh. Astronom., 7, 155–157.
Rychlik Z. (1983) Nonunifonn central limit bounds with applications to probability of deviations. Theory Probab. Appl, 28, 646–652.
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© 1994 Springer-Verlag Berlin Heidelberg
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Martikainen, A. (1994). One-Sided Deviations of a Random Walk Without Moment Assumptions. In: Mandl, P., Hušková, M. (eds) Asymptotic Statistics. Contributions to Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57984-4_33
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DOI: https://doi.org/10.1007/978-3-642-57984-4_33
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-7908-0770-7
Online ISBN: 978-3-642-57984-4
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