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sub-gaussian-random-variables-6.pdf

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OpenStax-CNX module: m37185 1 Sub-Gaussian random variables ∗ Mark A. Davenport This work is produced by OpenStax-CNX and licensed under the † Creative Commons Attribution License 3.0 Abstract In this module we introduce the sub-Gaussian and strictly sub-Gaussian distributions. We provide some simple examples and illustrate some of the key properties of sub-Gaussian random variables. A number of distributions, notably Gaussian and Bernoulli, are known to satisfy certain concentration of me
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        ∗      †                X         c >  0   E ( exp ( Xt )) ≤ exp  c 2 t 2 / 2     t ∈ R    X   ∼ Sub  c 2     X    E ( exp ( Xt ))        X          X   ∼ N   0 ,σ 2    X     σ 2   X   ∼ Sub  σ 2    E ( exp ( Xt )) =  exp  σ 2 t 2 / 2       X     B    | X  |≤ B    X   ∼ Sub  B 2         X   ∼ Sub  c 2   E ( X  ) = 0    E  X  2  ≤ c 2 .   ∗   †              X   ∼  Sub  c 2     E  X  2   ≤  c 2        X         X   ∼ Sub  σ 2     σ 2 =  E  X  2     E ( exp ( Xt )) ≤ exp  σ 2 t 2 / 2     t  ∈  R    X     σ 2    X   ∼ SSub  σ 2       X   ∼N   0 ,σ 2     X   ∼ SSub  σ 2       X   ∼ U   ( − 1 , 1)   X     [ − 1 , 1]    X   ∼ SSub(1 / 3)      P ( X   = 1) =  P ( X   = − 1) = 1 − s 2  ,  P ( X   = 0) =  s, s ∈ [0 , 1) .    s ∈ [0 , 2 / 3]   X   ∼ SSub(1 − s )    s ∈ (2 / 3 , 1)   X        X   ∼  Sub  c 2     t 0  ≥  0    a  ≥  0    P ( | X  |≥ t ) ≤ 2 exp  −  t 2 2 a 2     t ≥ t 0    X   ∼ SSub  σ 2     t >  0    a  =  σ        X   = [ X  1 ,X  2 ,...,X  N  ]    X  i    X  i  ∼  Sub  c 2     α  ∈  R N   < X,α > ∼  Sub  c 2  α  22    X  i  ∼ SSub  σ 2     α ∈ R N   < X,α > ∼ SSub  σ 2  α  22       X  i  E  exp  t  N i =1  α i X  i   =  E   N   i =1 exp ( tα i X  i )  = N   i =1 E ( exp ( tα i X  i )) ≤ N   i =1 exp  c 2 ( α i t ) 2 / 2  =  exp  N i =1  α 2 i  c 2 t 2 / 2  .          X  i    c 2 =  σ 2   E  < X,α > 2   =  σ 2  α  22         

Slide 3

Jul 23, 2017

Paper293886.pdf

Jul 23, 2017
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