Comparison of moments of sums of independent random variables and differential inequalities
The paper has two parts: I. Poincaré type inequalities and comparison of \(2p\), \(p\) moments \(\mu_r=(\int|\;|^r)^{1/r}\) of \(S=\sum x_i\xi_i\), \(\xi_i\) independent and symmetric, \(x_i\) in a normed space, and II. Logarithmic Sobolev type inequalities and comparison of \(p>q>1\) moments of \(S\). In I, for every probability \(M\) having a continuous positive symmetric density \(m\) on \(I=(-a,a)\), \(a\) possibly infinite, the authors define a number \(\lambda_2(M)\) as the infimum of all \(\lambda>1\) in the spectrum of an operator \(L\) on \(L^2(M)\), defined roughly as \(xf'(x)-u(x)f''(x)\), where \(u=p/m\), and \(p(x)=\int^a_x sm(s)ds\). Then, if \(\lambda_2\) of all \(\xi_i\) are \(>\lambda\), \(\lambda\in(1,2]\), an inequality \(\mu_{2p}\leq C_{p,\lambda}\mu_p\) is proved, for \(1\leq p\leq b_\lambda\), \(b_\lambda\) and \(C_{p,\lambda}\) having explicit simple expressions. As examples normal, uniform, symmetric beta, symmetric exponential and also a corollary for symmetric Bernoulli are given. In II, \(k(M)\) is defined as the maximal \(c\) for which, whatever be \(f\in C^\infty_0(I)\), \[ \int f^2\ln f^2dM\leq\Bigl(\int f^2dM\Bigr)\ln\Bigl(\int f^2dM\Bigr)+(4/c)\int f Lf dM \] holds. Then, if \(k(\xi_i)\geq k\), it is proved that the \(q\) moment of \(a \sum v_{i_1\cdots i_r}\xi_{i_1}\cdots\xi_{i_r}\), where \(r\) runs over \(0,\dots,n\) and \(i_j\) is increasing in \(j\) and between 1 and \(n\), is not smaller than the \(p\) moment of an analogous sum, but with explicitly expressed (simple) coefficients \(\sigma^r_{p,q,k}\). As examples normal, symmetric \(\text{gamma}_{\alpha,\beta}\), Weibull. We have also to remark that the paper is written according to ``we give two methods'' in the introduction, i.e. it contains discussions, ideas for shortening some known proofs etc.
scientific article; zbMATH DE number 871746