A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables

We introduce a Bernstein-type inequality which serves to uniformly control quadratic forms of gaussian variables. The latter can for example be used to derive sharp model selection criteria for linear estimation in linear regression and linear inverse problems via penalization, and we do not exclude that its scope of application can be made even broader.

💡 Research Summary

The paper introduces a novel Bernstein‑type concentration inequality specifically tailored for quadratic forms of Gaussian random variables. While classical Bernstein inequalities provide sharp tail bounds for linear combinations of independent sub‑Gaussian variables, they do not directly address the more complex structure of quadratic forms (Q = X^{\top} A X), where (X) is a centered Gaussian vector with covariance matrix (\Sigma) and (A) is a symmetric matrix. This gap is significant because many statistical and signal‑processing problems—such as risk assessment in linear regression, regularization in inverse problems, and model‑selection criteria—naturally involve such quadratic expressions.

Key Contributions

1. Derivation of a Uniform Tail Bound
   By diagonalising the symmetric matrix (A = U \Lambda U^{\top}) and rotating the Gaussian vector to (Y = U^{\top} X), the quadratic form decomposes into a weighted sum of independent (\chi^{2}{1}) variables: (Q = \sum{k=1}^{n} \lambda_{k} Y_{k}^{2}). The authors then apply a refined version of the classical Bernstein inequality to each term, carefully handling the non‑identical weights (\lambda_{k}). Using a martingale exponential super‑martingale construction together with the cumulant generating function of a centered (\chi^{2}_{1}) variable, they obtain the following concentration inequality for all (t \ge 0): \