Finite-Order Hilbertian Gaussian Random Tensor Estimates

Abstract

We prove fixed finite-chaos-order estimates for Hilbert-space-valued Gaussian random tensors. Given a finite-rank kernel \[ K\in\cA_1\otimes\cdots\otimes\cA_m\otimes\cC\otimes\cE \] and the associated decoupled homogeneous Gaussian chaos operator $\cT_K^{(m)}:\cC\to\cE$, we show that, for $p\ge2$ and $2\le r<\infty$, \[ \|\cT_K^{(m)}\|_{L^p(Ω;\mathfrak S_r(\cC,\cE))} \le C_m(p+r)^{m/2} \max_{S\subset[m]}\|\cF_S(K)\|_{\mathfrak S_r}, \] where $\cF_S(K):\cA_S\otimes\cC\to\cA_{S^c}\otimes\cE$ is the oriented input-output flattening. The proof is an induction on $m$ from the rectangular non-commutative Khintchine inequality: the two square functions place the last stochastic leg on the input or output side, producing all oriented flattenings. We also derive operator-norm, rank-logarithmic, tail, Borel--Cantelli cutoff-Cauchy, same-field Wick-chaos, binary Wick-product, and completion consequences. The estimates provide deterministic flattening certificates for random operator bounds in finite Gaussian/Wick expansions. Constants depend only on the fixed chaos order and not on Hilbert-space dimensions or cutoff ranks. Thus finite order means finitely many stochastic legs, not finite-dimensional Hilbert spaces; finite-rank kernels are model cutoffs, and the infinite-dimensional statement is obtained by completion in the maximum oriented Schatten-flattening norm.