Deviation probability bound for martingales with applications to statistical estimation (Q1970829)

The authors present an upper bound for the probability that \[ z^*\langle M\rangle_t^{-1}M_t>\lambda\sqrt{z^*\langle M\rangle_t^{-1}z} \] with a fixed vector \(z\) and \(\lambda\geq 1\). Here \(M_t\) is a vector martingale and \(\langle M\rangle_t\) denotes its predictable quadratic variation. They discuss some of its applications to statistical estimation in autoregressive and linear diffusion models. Their approach is non-asymptotic and does not require any ergodic assumption on the underlying model.