# Limit Theorems for Stopped Random Walks

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

## Abstract

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.

## Keywords

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.

## Preview

1. 23.
Anscombe, F.J. (1952): Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48, 600-607.
2. 35.
Baum, L.E. and Katz, M. (1965): Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108-123.
3. 38.
Billingsley, P. (1968): Convergence of Probability Measures. John Wiley, New York.
4. 44.
Blackwell, D. (1946): On an equation of Wald. Ann. Math. Statist. 17, 84-87.
5. 46.
Blackwell, D. (1953): Extension of a renewal theorem. Pacific J. Math. 3, 315-320.
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.
8. 68.
Chow, Y.S., Robbins, H. and Siegmund, D. (1971): Great Expectations: The Theory of Optimal Stopping. Houghton-Miffiin, Boston, MA.
9. 69.
Chow, Y.S., Robbins, H. and Teicher, H. (1965): Moments of randomly stopped sums. Ann. Math. Statist. 36, 789-799.
10. 73.
Chung, K.L. (1974): A Course in Probability Theory, 2nd ed. Academic Press, New York.
11. 89.
De Groot, M.H. (1986): A conversation with David Blackwell. Statistical Science 1, 40-53.
12. 103.
Erdős, P. (1949): On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286-291.
13. 104.
Erdős, P. (1950): Remark on my paper “On a theorem of Hsu and Robbins.” Ann. Math. Statist. 21, 138.
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.
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.
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.
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.
21. 156.
Hartman, P. and Wintner, A. (1941): On the law of the iterated logarithm. Amer. J. Math. 63, 169-176.
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.
23. 182.
Katz, M.L. (1963): The probability in the tail of a distribution. Ann. Math. Statist. 34, 312-318.
24. 192.
Lai, T.L. (1975): On uniform integrability in renewal theory. Bull Inst. Math. Acad. Sinica 3, 99-105.
25. 216.
Loéve, M. (1977): Probability Theory, 4th ed. Springer-Verlag, New York.
26. 228.
Neveu, J. (1975): Discrete-Parameter Martingales. North-Holland, Amsterdam.
27. 240.
Pyke, R. and Root, D. (1968): On convergence in r-mean for normalized partial sums. Ann. Math. Statist. 39, 379-381.
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.
29. 244.
Richter, W. (1965): Limit theorems for sequences of random variables with sequences of random indices. Theory Probab. Appl. X, 74-84.
30. 258.
Smith, W.L. (1955): Regenerative stochastic processes. Proc. Roy. Soc. London Ser. A 232, 6-31.
31. 290.
Stout, W.F. (1974): Almost Sure Convergence. Academic Press, New York.
32. 292.
Strassen, V. (1966): A converse to the law of the iterated logarithm. Z. Wahrsch. verw. Gebiete 4, 265-268.
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.
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

## Authors and Affiliations

1. 1.Department of MathematicsUppsala UniversityUppsalaSweden