PAC-Bayesian-Like Error Bound for a Class of Linear Time-Invariant Stochastic State-Space Models

The paper derives KL- and Rényi-type PAC-Bayesian-like generalization bounds for stochastic LTI predictors (Theorems 4.1 and 4.2) via (i) replacing the empirical loss by an infinite-past proxy VN(f), (ii) bounding VN(f)−L̂N(f) by a transient term 2(δN)−1Eρ G(f), and (iii) controlling L(f)−VN(f) using either a KL change-of-measure with an mgf bound or a Rényi/Hölder step with high-order moment bounds. The candidate solution reproduces the same three pillars: the linear-Gaussian error filter (Definition 4.1), the transient control via G(·), and the fluctuation control yielding the same constants and ranges (including the (m+r−1)!(r−1) combinatorics and the denominator 1−3(m+1)λμmax(Qe)Ge(·)2). Minor presentational differences (e.g., an explicit combinatorial inequality and a brief mention of Wick/Isserlis) do not alter the substance. The final KL and Rényi bounds match those stated in the paper, up to benign constant slack already present in the paper’s own appendix. Therefore, both are correct, and the proofs are substantially the same. Key correspondences: Definition 4.1 and the error filter representation; Corollary B.1 for the 2/(δN) transient term; Lemma B.8/B.10 for the moment/mgf bounds; and Theorems 4.1–4.2 for the final forms and constants.