Summary
LetX 1,X 2, ... be a sequence of independent random variables with common lattice distribution functionF having zero mean, and let (S n ) be the random walk of partial sums. The strong law of large numbers (SLLN) implies that for any α∈ℝ and ε>0
decreases to 0 asm increases to ∞. Under conditions on the moment generating function ofF, we obtain the convergence rate by determiningp m up to asymptotic equivalence. When α=0 and ε is a point in the lattice forF, the result is due to Siegmund [Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 107–113 (1975); but this restriction on ε precludes all small values of ε, and these values are the most interesting vis-à-vis the SLLN. Even when α=0 our result handles important distributionsF for which Siegmund's result is vacuous, for example, the two-point distributionF giving rise to simple symmetric random walk on the integers. We also identify for both lattice and non-lattice distributions the behavior of certain quantities in the asymptotic expression forp m as ε decreases to 0.
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Bahadur, R.R., Rao, R. Ranga: On deviations of the sample mean. Ann. Math. Stat.31, 1015–1027 (1960)
Blackwell, d., Hodges, J.L., Jr.: The probability in the extreme tail of a convolution. Ann. Math. Stat.30, 1113–1120 (1959)
Chung, K.L.: A course in probability theory, 2nd ed. New York: Academic Press 1974
Daniels, H.E.: Tail probability approximations. Int. Stat. Rev.55, 37–48 (1987)
Feller, W.: An introduction to probability theory and its applications, Vol. 2, 2nd ed. New York: Wiley 1971
Fill, J.A.: convergence rates related to the strong law of large numbers. Stanford University Technical Report No. 14 (1980)
Fill, J.A.: Convergence rates related to the strong law of large numbers. Ann. Probab.11, 123–143 (1983)
Fill, J.A.: Asymptotic expansions for large deviation probabilities in the strong law of large numbers. Probab. Th. Rel. Fields81, 213–233 (1989)
Lai, T.Z.: Asymptotic moments of random walks with applications to ladder variables and renewal theory. Ann. Probab.4, 51–66 (1976)
Lugannani, R., Rice, S.: Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab.12, 475–490 (1980)
Müller, D.W.: Verteilungs-Invarianzprinzipien für das Gesetz der großen Zahlen. Z. Wahrscheinlichkeitstheor. Verw. Geb.10, 173–192 (1968)
Siegmund, D.: Large deviation probabilities in the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 107–113 (1975)
Siegmund, D.: Sequential analysis. New York Berlin Heidelberg: Springer 1985
Spitzer, F.: A Tauberian theorem and its probability interpretation. Trans. Am. Math. Soc.94, 150–169 (1960)
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The work of this author was carried out in part while at Stanford University and in part while on leave at the University of Chicago
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Fill, J.A., Wichura, M.J. The convergence rate for the strong law of large numbers: General lattice distributions. Probab. Th. Rel. Fields 81, 189–212 (1989). https://doi.org/10.1007/BF00319550
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DOI: https://doi.org/10.1007/BF00319550