# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## 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 $( \max ( 1,\ln p) ) ^{d^2}.$

#### Autorzy

• Rafał M. ŁochowskiInstitute of Mathematics
Warsaw University
Banacha 2
02-097 Warszawa, Poland
and
Department of Mathematical Economics
Warsaw School of Economics
Al. Niepodległości 164
02-554 Warszawa, Poland
e-mail

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