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Cramér-Wold theorem


Let

ˉXn=(Xn1,,Xnk)andˉX=(X1,,Xk)

be random vectors. Then ˉXn converges to ˉX in distributionPlanetmathPlanetmath (http://planetmath.org/ConvergenceInDistribution) if and only if

ki=1tiXni𝐷nki=1tiXi.

for each (t1,,tk)k. That is, if every linear combinationMathworldPlanetmath of the coordinatesPlanetmathPlanetmath of ˉXn converges in distribution to the correspondent linear combination of coordinates of ˉX.

Title Cramér-Wold theorem
Canonical name CramerWoldTheorem
Date of creation 2013-03-22 13:14:21
Last modified on 2013-03-22 13:14:21
Owner Koro (127)
Last modified by Koro (127)
Numerical id 4
Author Koro (127)
Entry type Theorem
Classification msc 60E05