Self-normalised Bennett inequalities for Hilbert-valued martingales

Abstract

We prove time-uniform self-normalised Bennett inequalities for a martingale $(M_n)_{n\geq0}$ in a separable Hilbert space, with $M_0=0$ and increments bounded in norm by one. Writing $V_n$ for its predictable covariance process and $h(u)=(1+u)\log(1+u)-u$ for the Bennett rate function, we show that, for every regularisation parameter $\rho>0$, the process \[\exp\left\{\rho\,h\!\left(\frac{\left\lVert (V_n+\rho I)^{-1/2}M_n\right\rVert}{\sqrt{\rho}}\right)-\frac12\log\det\left(I+\rho^{-1}V_n\right)\right\},\qquad n\geq0,\] is a nonnegative supermartingale with initial value one, where $\det$ is the Fredholm determinant. Ville's inequality yields time-uniform Bennett and Bernstein bounds for $\lVert (V_n+\rho I)^{-1/2}M_n\rVert$. The result permits conditional covariance increments of infinite rank; in finite dimensions, the resulting boundaries sharpen existing martingale-transform and determinant-based variational bounds. The same construction extends to compensated marked point processes with bounded jumps. Mixing these supermartingales over $\rho$ gives simultaneous control over the regularisation parameter. Consequences include an upper law of the iterated logarithm for the regularised ellipsoidal radius in separable Hilbert spaces and, in finite dimensions, spectrum-sensitive finite-time bounds and an upper law of the iterated logarithm for the unregularised radius $\lVert V_n^{-1/2}M_n\rVert$, whose constant $1$ is sharp over the class. We also obtain a time-uniform Bernstein inequality for the martingale norm $\lVert M_n\rVert$ with dependence on $\operatorname{tr}(V_n)$ and $\lVert V_n\rVert_{\mathrm{op}}$.