Let h:[0,1]→R be C² and such that sup_[0,1] h'' < 0. For a (large) positive integer n, set h_n(k) = n h(k/n) for any k in {0,...,n}. We consider a random walk (S_k)_{k\geq 0} with i.i.d. centred increments having some finite exponential moments. We are interested in the event {S ≥ h_n} = {S_k ≥ h_n(k) for all k in {0,...,n}}. It is well known that P(S ≥ h_n | S₀=0, S_n= [h_n(n)]) = exp(-n ∫_[0,1] I(h'(s)) ds + o(n)), where I is the Legendre-Fenchel transform of the log-moment generating function associated to the increments. We first prove that the leading correction is of order exp(-Θ(n^1/3)). We then turn our attention to the conditional random walk measure P^h_n = P( · | S ≥ h_n, S₀=0, S_n=[h_n(n)]). We prove that the one-point tails are of the form P^h_n (S_k ≥ h_n(k) + t n^1/3) = exp(-Θ(t^3/2)) for all t<n^β for any β in (0,1/6). Moreover, we prove that, for any r ≥ 1, E^h_n((S_k-h_n(k))^r) = Θ(n^r/3) and Var^h_n(S_k) = Θ(n^2/3), for all k far enough from 0 and n. In addition, we show that Cov^h_n(S_k,S_l) ≤ O(n^2/3) exp(-O(|l-k|/n^2/3)) for all k,l not too close to 0 and n.