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单词 MultidimensionalChebyshevsInequality
释义

Multidimensional Chebyshev’s inequality


Let X be an N-dimensional random variableMathworldPlanetmath with mean μ=𝔼[X] and covariance matrixMathworldPlanetmath V=𝔼[(X-μ)(X-μ)T].

If V is invertiblePlanetmathPlanetmath (i.e., strictly positive), for any t>0:

Pr((X-μ)TV-1(X-μ)>t)Nt2

Proof:V is positive, so V-1 is.Define the random variable

y=(X-μ)TV-1(X-μ)

y is positive, then Markov’s inequality holds:

Pr((X-μ)TV-1(X-μ)>t)=Pr(y>t)=Pr(y>t2)𝔼[y]t2

Since V is symmetricPlanetmathPlanetmath, a rotationMathworldPlanetmath R (i.e., RRT=RTR=I) and a diagonal matrixMathworldPlanetmath D (i.e., ijDi,j=0) exist such that

V=RTDR

Since V is positive Dii>0.Besides

V-1=R-1D-1(RT)-1=RTD-1R

clearly [D-1]ii=1Dii.

Define Z=R(X-μ).

The following identities hold:

𝔼[ZZT]=R𝔼[(X-μ)(X-μ)T]RT=RRTDRRT=Di𝔼[Zi2]=Dii

and

y=ZTRV-1RTZ=ZTD-1Z=i=1NZi2Dii

then

𝔼[y]=i=1N𝔼[Zi2]Dii=N
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更新时间:2025/5/4 17:21:14