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单词 BernsteinInequalities
释义

Bernstein inequalities


1) Let {Xi}i=1n be a collectionMathworldPlanetmath of independentPlanetmathPlanetmath randomvariablesMathworldPlanetmath satisfying the conditions:
a) E[Xi2]< i, so that one can write i=1nE[Xi2]=v2
b) c:i=1nE[|Xi|k]12k!v2ck-2 for all integers k3

Then, for any ε0,

Pr{i=1n(Xi-E[Xi])>ε}exp[-v2c2(1+cεv2-1+2cεv2)]exp(-ε22(v2+cε))
Pr{|i=1n(Xi-E[Xi])|>ε}2exp[-v2c2(1+cεv2-1+2cεv2)]2exp(-ε22(v2+cε))

2) Let {Xi}i=1n be a collection of independent, almostsurely absolutely bounded (http://planetmath.org/AlmostSurelyBoundedRandomVariable) random variables, that is Pr{|Xi|M}=1 i.
Then, for any ε0,

Pr{i=1n(Xi-E[Xi])>ε}exp[-9v2M2(1+Mε3v2-1+2Mε3v2)]exp(-ε22(v2+M3ε))
Pr{|i=1n(Xi-E[Xi])|>ε}2exp[-9v2M2(1+Mε3v2-1+2Mε3v2)]2exp(-ε22(v2+M3ε))
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更新时间:2025/5/4 16:41:06