Self-normalised Bennett inequalities for Hilbert-valued martingales

Abstract

We prove time-uniform self-normalised Bennett inequalities for a martingale $(M_n{)}_{n\ge 0}$ 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 \[\exp\left\{ρ\,h\!\left(\frac{\left\lVert (V_n+ρI)^{-1/2}M_n\right\rVert}{\sqrtρ}\right)-\frac12\log\det\left(I+ρ^{-1}V_n\right)\right\},\qquad n\geq0,\] 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 $\Vert (V_n+\rho I{)}^{-1/2}{M}_n\Vert$. 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 $\Vert V_n^{-1/2}{M}_n\Vert$, whose constant $1$ is sharp over the class. We also obtain a time-uniform Bernstein inequality for the martingale norm $\Vert M_n\Vert$ with dependence on $\mathrm{tr}(V_n)$ and $\Vert V_n{\Vert }_{\mathrm{op}}$.