Summary
We obtain a general Darling-Erdős type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erdős theorem holds in its classical formulation if and only ifE[X 2 1 {|X|≧t}]=o((loglogt)-1) ast→∞. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.
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Einmahl, U. The Darling-Erdős theorem for sums of I.I.D. random variables. Probab. Th. Rel. Fields 82, 241–257 (1989). https://doi.org/10.1007/BF00354762
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DOI: https://doi.org/10.1007/BF00354762
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Classical Formulation
- Type Theorem