Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings
HTML articles powered by AMS MathViewer
- by Y. S. Chow and T. L. Lai PDF
- Trans. Amer. Math. Soc. 208 (1975), 51-72 Request permission
Abstract:
In this paper, we prove certain one-sided Paley-type inequalities and use them to study the convergence rates for the tail probabilities of sample sums. We then apply our results to find the limiting moments and the limiting distribution of the last time and the largest excess of boundary crossings for the sample sums, generalizing the results previously obtained by Robbins, Siegmund and Wendel. Certain one-sided limit theorems for delayed sums are also obtained and are applied to study the convergence rates of tail probabilities.References
- Leonard E. Baum and Melvin Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108–123. MR 198524, DOI 10.1090/S0002-9947-1965-0198524-1
- Y. S. Chow, Delayed sums and Borel summability of independent, identically distributed random variables, Bull. Inst. Math. Acad. Sinica 1 (1973), no. 2, 207–220. MR 343357
- Y. S. Chow, Herbert Robbins, and David Siegmund, Great expectations: the theory of optimal stopping, Houghton Mifflin Co., Boston, Mass., 1971. MR 0331675
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- P. Erdös, On a theorem of Hsu and Robbins, Ann. Math. Statistics 20 (1949), 286–291. MR 30714, DOI 10.1214/aoms/1177730037
- Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1–125. MR 14626, DOI 10.1007/BF02392223
- Philip Hartman and Aurel Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169–176. MR 3497, DOI 10.2307/2371287 C. S. Kao, On the time and the excess of linear boundary crossings of the sample sums, Ph.D. Thesis, Columbia University, New York, 1972.
- J. Kiefer and J. Wolfowitz, On the characteristics of the general queueing process, with applications to random walk, Ann. Math. Statist. 27 (1956), 147–161. MR 77019, DOI 10.1214/aoms/1177728354
- J. F. C. Kingman, Some inequalities for the queue $GI/G/1$, Biometrika 49 (1962), 315–324. MR 198565, DOI 10.1093/biomet/49.3-4.315
- Tze Leung Lai, Limit theorems for delayed sums, Ann. Probability 2 (1974), 432–440. MR 356193, DOI 10.1214/aop/1176996658
- Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748 J. Marcinkiewicz et A. Zygmund, Quelques théorèmes sur les fonctions indépendantes, Studia Math. 7 (1938), 104-120.
- Herbert Robbins and David Siegmund, Boundary crossing probabilities for the Wiener process and sample sums, Ann. Math. Statist. 41 (1970), 1410–1429. MR 277059, DOI 10.1214/aoms/1177696787
- H. Robbins, D. Siegmund, and J. Wendel, The limiting distribution of the last time $s_{n}\geq n\varepsilon$, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1228–1230. MR 243625, DOI 10.1073/pnas.61.4.1228
- Frank Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–339. MR 79851, DOI 10.1090/S0002-9947-1956-0079851-X
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
- Leonard E. Baum, On convergence to $+\infty$ in the law of large numbers, Ann. Math. Statist. 34 (1963), 219–222. MR 143240, DOI 10.1214/aoms/1177704258
- D. W. Müller, Verteilungs-Invarianzprinzipien für das starke Gesetz der grossen Zahl, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 173–192 (German, with English summary). MR 232428, DOI 10.1007/BF00531847
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 208 (1975), 51-72
- MSC: Primary 60G50
- DOI: https://doi.org/10.1090/S0002-9947-1975-0380973-6
- MathSciNet review: 0380973