Abstract
Let \(\{X_i, i=1, 2, 3, \ldots\}\) be a sequence of independent and identically distributed random variables and write \(S_n=\sum\nolimits^n_{i=1} X_i,\: n\geqq1.\) It is well known that
If and only if \(EX_i=0,\: EX^2_i=1\) (Hartman and Wintner [6] obtained the sufficiency part and Strassen [9] the necessity) and the purpose of this note is to clarify the corresponding situation in the case where these moment conditions are violated. It turns out that the oscillation behavior of normed sums rests essentially on whether or not the summands belong to the domain of partial attraction of the normal distribution.
Received by the editors November 1, 1968.
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References
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Heyde, C.C. (2010). A Note Concerning Behaviour of Iterated Logarithm Type. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_15
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