Moment estimates for chaoses with regular moment growth

Abstract

We give two-sided moment estimates for decoupled chaoses of any order generated by independent symmetric random variables with regular moment growth: $\|X\|_{2p}\le K\|X\|_p$ for every $p\ge1$. The estimates involve only the coefficient tensor and the one-dimensional tails. They use the partition norms of Adamczak and Lata{\l}a, with constants depending only on the order and on $K$. The proof starts with log-concave tails. Gaussian smoothing gives the estimate for the expected norm needed for induction, and a product comparison reduces the remaining distributions to this case. The extra tensor indices introduced by the comparison are removed using Rademacher moments. A comparison with products of log-concave-tail variables then extends the formula to regular moment growth. We also give tail bounds and explicit Rademacher and Weibull formulas for $r\ge1$, including fourth-order examples. This article was developed with the assistance of GPT Astra-6.