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Journal of the London Mathematical Society 1981 s2-24(3):562-568; doi:10.1112/jlms/s2-24.3.562
© 1981 by London Mathematical Society
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© Oxford University Press

Order of Magnitude of Moments of Sums of Random Variables

Peter Hall

Department of Mathematics, Australian National University P.O. Box 4, Canberra, A.C.T., Australia 2600

Let X, X1, X2, ... be a sequence of independent and identically distributed random variables, set Sn = Formula and let med Sn denote the median of Sn. Suppose that 0 < p ≤ 2 and E|X|P < {infty}. We derive the precise order of magnitude of E|Sn – med Sn|p by obtaining a sequence of constants {lambda}p(n) which depends on the distribution of X in a very simple way and which satisfies C1{lambda}p(n) ≤ E|Sn – med Sn|p ≤ C2{lambda}p(n) for positive constants C1 and C2 not depending on n. We obtain similar bounds for centring constants other than the median, and compare them.


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