Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails
Tom 72 / 2006
Banach Center Publications 72 (2006), 161-176
MSC: Primary 60E15.
DOI: 10.4064/bc72-0-11
Streszczenie
Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S=\sum a_{i_{1},\dots ,i_{d}}X_{i_{1}}^{( 1) }\cdots X_{i_{d}}^{( d) }$ generated by symmetric random variables $X_{i_{1}}^{( 1) },\dots , X_{i_{d}}^{( d) }$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order $d.$ Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of $S$ and moments of some related Rademacher chaoses. The estimates for $p$th moment are exact up to a factor $( \max ( 1,\ln p) ) ^{d^2}.$
