Summary
LetX 1,X 2,... be a sequence of independent random variables with common distribution functionF having zero mean, and let (S n ) be the random walk of partial sums. The weak and strong laws of large numbers, respectively, imply that for any α∈ℝ and ε>0 the probabilitiesP{S m >α+εm} and
tend to 0 asm tends to ∞. Building upon work of Bahadur and Ranga Rao [Ann. Math. Stat.31, 1015–1027 (1960)], Siegmund [Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 107–113 (1975)], and Fill and Wichura [Probab. Th. Rel. Fields81, 189–212 (1989)], we produce complete asymptotic expansions for the probabilitiesP{S m >α+εm} andp m .
References
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Written while the author was on leave from Stanford University at the University of Chicago
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Fill, J.A. Asymptotic expansions for large deviation probabilities in the strong law of large numbers. Probab. Th. Rel. Fields 81, 213–233 (1989). https://doi.org/10.1007/BF00319551
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DOI: https://doi.org/10.1007/BF00319551
Keywords
- Stochastic Process
- Random Walk
- Probability Theory
- Asymptotic Expansion
- Mathematical Biology