Limit Theorems for Stopped Random Walks

Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)


Classical limit theorems such as the law of large numbers, the central limit theorem and the law of the iterated logarithm are statements concerning sums of independent and identically distributed random variables, and thus, statements concerning random walks. Frequently, however, one considers random walks evaluated after a random number of steps. In sequential analysis, for example, one considers the time points when the random walk leaves some given finite interval. In renewal theory one considers the time points generated by the so-called renewal counting process. For random walks on the whole real line one studies first passage times across horizontal levels, here, in particular, the zero level corresponds to the first ascending ladder epoch. In reliability theory one may, for example, be interested in the total cost for the replacements made during a fixed time interval and so on.


Random Walk Limit Theorem Central Limit Theorem Iterate Logarithm Complete Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 23.
    Anscombe, F.J. (1952): Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48, 600-607.MATHCrossRefMathSciNetGoogle Scholar
  2. 35.
    Baum, L.E. and Katz, M. (1965): Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108-123.MATHCrossRefMathSciNetGoogle Scholar
  3. 38.
    Billingsley, P. (1968): Convergence of Probability Measures. John Wiley, New York.MATHGoogle Scholar
  4. 44.
    Blackwell, D. (1946): On an equation of Wald. Ann. Math. Statist. 17, 84-87.CrossRefMathSciNetGoogle Scholar
  5. 46.
    Blackwell, D. (1953): Extension of a renewal theorem. Pacific J. Math. 3, 315-320.MATHMathSciNetGoogle Scholar
  6. 57.
    Chang, I. and Hsiung, C (1979): On the uniform integrability of \(| b-^{1/p} W_{Mb} | ^{p},0 < V <\) 2. Preprint, NCU Chung-Li, Taiwan.Google Scholar
  7. 63.
    Chow, Y.S., Hsiung, C.A. and Lai, T.L. (1979): Extended renewal theory and moment convergence in Anscombe’s theorem. Ann. Probab. 7, 304-318.MATHCrossRefMathSciNetGoogle Scholar
  8. 68.
    Chow, Y.S., Robbins, H. and Siegmund, D. (1971): Great Expectations: The Theory of Optimal Stopping. Houghton-Miffiin, Boston, MA.MATHGoogle Scholar
  9. 69.
    Chow, Y.S., Robbins, H. and Teicher, H. (1965): Moments of randomly stopped sums. Ann. Math. Statist. 36, 789-799.MATHCrossRefMathSciNetGoogle Scholar
  10. 73.
    Chung, K.L. (1974): A Course in Probability Theory, 2nd ed. Academic Press, New York.MATHGoogle Scholar
  11. 89.
    De Groot, M.H. (1986): A conversation with David Blackwell. Statistical Science 1, 40-53.CrossRefMathSciNetGoogle Scholar
  12. 103.
    Erdős, P. (1949): On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286-291.CrossRefGoogle Scholar
  13. 104.
    Erdős, P. (1950): Remark on my paper “On a theorem of Hsu and Robbins.” Ann. Math. Statist. 21, 138.CrossRefGoogle Scholar
  14. 114.
    Feller, W. (1968): An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. John Wiley, New York.Google Scholar
  15. 131.
    Gut, A. (1974a): On the moments and limit distributions of some first passage times. Ann. Probab. 2, 277-308.MATHCrossRefMathSciNetGoogle Scholar
  16. 132.
    Gut, A. (1974b): On the moments of some first passage times for sums of dependent random variables. Stoch. Process. Appl. 2, 115-126.MATHCrossRefMathSciNetGoogle Scholar
  17. 133.
    Gut, A. (1974c): On convergence in r-mean of some first passage times and randomly indexed partial sums. Ann. Probab. 2, 321-323.MATHCrossRefMathSciNetGoogle Scholar
  18. 137.
    Gut, A. (1983b): Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times. Acta Math. Acad. Sci. Hungar. 42, 225-232; Correction, ibid. 45 (1985), 235-236.Google Scholar
  19. 145.
    Gut, A. (2007): Probability: A Graduate Course, Corr. 2nd printing. Springer-Verlag, New York.Google Scholar
  20. 149.
    Gut, A. and Janson, S. (1986): Converse results for existence of moments and uniform integrability for stopped random walks. Ann. Probab. 14, 1296-1317.MATHCrossRefMathSciNetGoogle Scholar
  21. 156.
    Hartman, P. and Wintner, A. (1941): On the law of the iterated logarithm. Amer. J. Math. 63, 169-176.CrossRefMathSciNetGoogle Scholar
  22. 169.
    Hsu, P.L. and Robbins, H. (1947): Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33, 25-31.MATHCrossRefMathSciNetGoogle Scholar
  23. 182.
    Katz, M.L. (1963): The probability in the tail of a distribution. Ann. Math. Statist. 34, 312-318.MATHCrossRefMathSciNetGoogle Scholar
  24. 192.
    Lai, T.L. (1975): On uniform integrability in renewal theory. Bull Inst. Math. Acad. Sinica 3, 99-105.MATHMathSciNetGoogle Scholar
  25. 216.
    Loéve, M. (1977): Probability Theory, 4th ed. Springer-Verlag, New York.MATHGoogle Scholar
  26. 228.
    Neveu, J. (1975): Discrete-Parameter Martingales. North-Holland, Amsterdam.MATHGoogle Scholar
  27. 240.
    Pyke, R. and Root, D. (1968): On convergence in r-mean for normalized partial sums. Ann. Math. Statist. 39, 379-381.MATHCrossRefMathSciNetGoogle Scholar
  28. 241.
    Rényi, A. (1957): On the asymptotic distribution of the sum of a random number of independent random variables. Ada Math. Acad. Sci. Hungar. 8, 193-199.MATHCrossRefGoogle Scholar
  29. 244.
    Richter, W. (1965): Limit theorems for sequences of random variables with sequences of random indices. Theory Probab. Appl. X, 74-84.CrossRefGoogle Scholar
  30. 258.
    Smith, W.L. (1955): Regenerative stochastic processes. Proc. Roy. Soc. London Ser. A 232, 6-31.MATHCrossRefMathSciNetGoogle Scholar
  31. 290.
    Stout, W.F. (1974): Almost Sure Convergence. Academic Press, New York.MATHGoogle Scholar
  32. 292.
    Strassen, V. (1966): A converse to the law of the iterated logarithm. Z. Wahrsch. verw. Gebiete 4, 265-268.MATHCrossRefMathSciNetGoogle Scholar
  33. 297.
    Szynal, D. (1972): On almost complete convergence for the sum of a random number of independent random variables. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20, 571-574.MATHMathSciNetGoogle Scholar
  34. 322.
    Yu, K.F. (1979): On the uniform integrability of the normalized randomly stopped sums of independent random variables. Preprint, Yale University.Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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