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Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities

✍ Scribed by S. Kwapień; R. Latała; K. Oleszkiewicz

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
463 KB
Volume
136
Category
Article
ISSN
0022-1236

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✦ Synopsis

For S= x i ! i , where (! i ) is a sequence of independent, symmetric random variables and (x i ) is a sequence of vectors in a normed space we give two methods of proving inequalities (E &S& p ) 1Âp C p, q (E &S& q ) 1Âq with the constants C p, q independent of the sequence (x i ). The methods depend on using differential inequalities of Poincare or logarithmic Sobolev type. The obtained constants are usually better than the ones obtained by other methods. 1996 Academic Press, Inc. I. Poincare Type Inequalities and Comparison of 2p, p Moments of S

Let I=(&a, a), 0<a and let M be a probability distribution on I with density function m which is continuous, positive, and symmetric on I. We will assume that M has the finite second moment, i.e., I s 2 M(ds)= I s 2 m(s) ds< . Let p, q, u: I [ R be defined by p(x)= a x sm(s) ds, q(x)=

x 0 (1Âp(s)) ds, u(x)=p(x)Âm(x). The functions p, u are positive, continuous, and symmetric on I. Let C 0 (I) denote the class of smooth functions on I with compact supports contained in I. For f # C 0 (I) we define Lf (x)=xf $(x)&u(x) f "(x)= &(1Âm(x))( pf $)$ (x). The operator L is formally symmetric in L 2 (I, M) and we have

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## I. fntFoduction Let {X,,, n 2 1) be a sequence of independent random variables, P, and f, the distribution function and the characteristic fundion of the X,, respectively. Let us put SN = 2 X,, where N is a pasitive integer-valued random variable independent of X,, ?t 2 1. Furthermore, let { P,