Abstract
Suppose $S_n = \sum^n_1 X_j$, where $\{X_n\}$ is a sequence of random variables. Under progressively weaker hypotheses, Pyke and Root (1968), Chatterji (1969) and Chow (1971) have proved that $E|S_n - b_n|^r = o(n)$, where $0 < r < 2$ and $\{b_n\}$ is properly chosen. This paper gives a fairly elementary proof of Chow's result under further weakened hypotheses.
Citation
S. W. Dharmadhikari. M. Sreehari. "On Convergence in $r$-Mean of Normalized Partial Sums." Ann. Probab. 3 (6) 1023 - 1024, December, 1975. https://doi.org/10.1214/aop/1176996228
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