Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that $\left(\right|\left|Tⁿx\right||/\left|\right|Tⁿ\left|\right|)\notin {\ell }^q\left(\mathbb{N}\right)$, using techniques which involve the modulus of asymptotic uniform smoothness of X.