... 0. On the other hand, the first probability in the second line of (2.21) is bounded above by the probability of Btnot hittingi(DT2(x, ε)) = D(0,ε) during nεexcursions, each starting at ... δ=1(4.1)since the complementary upper bound on Tnis contained in [4, Cor. 25, Chap.7] (see also the references therein). Our approach is to use Theorem 1.2 to-gether with the strong approximation ... centered at x.Forx in M we have the ε-hitting timeT (x, ε) = inf{t>0 |Xt∈ DM(x, ε)}.ThenCε= supx∈MT (x, ε)is the ε-covering time of M.Proof of Theorem 1.3. If g denotes the Riemannian...