Sufficient Conditions for Existence of $J_{\alpha}(X + \sqrt[\alpha]{\eta}N)$

arXiv1604.02058MaRDI QIDQ6272302

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Publication date: 7 April 2016

Abstract: In his technical report~cite[sec. 6]{barrontech}, Barron states that the de Bruijn's identity for Gaussian perturbations holds for any RV having a finite variance. In this report, we follow Barron's steps as we prove the existence of

J_{a l p h a} l e f t (X + s q r t [a l p h a] e t a N i g h t)

,

e t a > 0

for any Radom Variable (RV)

X i n m a t h c a l L

where �egin{equation*} mathcal{L} = left{ ext{RVs} ,,U: int lnleft(1 + |U| ight),dF_{U}(u) ext{ is finite } ight}, end{equation*} and where

N s i m m a t h c a l S (a l p h a; 1)

is independent of

X

,

0 < a l p h a < 2

.

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