Normal characterization by zero correlations
2006 ◽
Vol 81
(3)
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pp. 351-361
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Keyword(s):
AbstractSuppose Xi, i = 1,…,n are indepedent and identically distributed with E/X1/r < ∞, r = 1,2,…. If Cov (( − μ)r, S2) = 0 for r = 1, 2,…, where μ = EX1, S2 = , and , then we show X1 ~ N (μ, σ2), where σ2 = Var(X1). This covariance zero condition charaterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of and S2 using semi invariants. More generally, if Cov = 0 for r = 1,…, k, then E((X1 − μ)/σ)r+2 = EZr+2 for r = 1,… k, where Z ~ N(0, 1). Conversely Corr may be arbitrarily close to unity in absolute value, but for unimodal X1, Corr2( < 15/16, and this bound is the best possible.
Keyword(s):
2001 ◽
Vol 51
(1-2)
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pp. 113-118
1971 ◽
Vol 1
(4)
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pp. 457-460
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1962 ◽
Vol 14
(1)
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pp. 173-178
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