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Order of Magnitude of Moments of Sums of Random Variables
Author(s) -
Hall Peter
Publication year - 1981
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-24.3.562
Subject(s) - sequence (biology) , mathematics , independent and identically distributed random variables , combinatorics , random variable , order (exchange) , magnitude (astronomy) , distribution (mathematics) , constant (computer programming) , set (abstract data type) , physics , mathematical analysis , statistics , chemistry , computer science , finance , astronomy , economics , biochemistry , programming language
Let X , X 1 , X 2 , … be a sequence of independent and identically distributed random variables, set S n =∑ 1 nX iand let med S n denote the median of S n . Suppose that 0 < p ⩽ 2 and E | X | P < ∞. We derive the precise order of magnitude of E | S n − med S n | p by obtaining a sequence of constants λ p ( n ) which depends on the distribution of X in a very simple way and which satisfies C 1 λ p ( n ) ⩽ E | S n − med S n | p ⩽ C 2 λ p ( n ) for positive constants C 1 and C 2 not depending on n . We obtain similar bounds for centring constants other than the median, and compare them.