Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails

Volume 72 / 2006

Rafał M. Łochowski Banach Center Publications 72 (2006), 161-176 MSC: Primary 60E15. DOI: 10.4064/bc72-0-11

Abstract

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

Authors

  • 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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