Tail and Moment Bounds for Gaussian Chaoses in Banach Spaces
summary
Provides two-sided Lp moment and tail bounds for homogeneous Gaussian chaoses of order d with Banach-space–valued coefficients, showing the Lp norm expands as a polynomial in sqrt(p): sum_{k=1}^d p^{k/2}·A_k (A_k are operator-like norms indexed by partitions of {1..d}), which yields Gaussian tails then exponential then heavier regimes. Proof uses decoupling, metric entropy and generic chaining (Dudley/Sudakov estimates) and gives a vector-valued Hanson–Wright extension; constants depend badly on d but improve to clean deterministic bounds in type-2 and L_r spaces.
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chapters (10)
- 0:00Start
- 4:32Introduction
- 5:53Symmetry conditions
- 7:02Moments vs tails
- 11:48Vector-valued linear forms d = 1, arbitrary
- 16:01Borell-Arcones-Giné estimate Application of the Gaussian concentration yields Theorem (Borell'84, Arcones-Giné'93)
- 19:54Easy lower bound
- 42:45Bounds for vector-valued chaoses of order 3
- 47:44Crucial tool - decoupling
- 55:04Upper bound for moments