Random Walk conditioned to stay above a non-flat floor: curvature effects

Abstract

Let $h:[0,1]\to \mathbb{R}$ be $C^2$ and such that ${\sup }_{[0,1]}{h}^{\prime \prime }<0$. For a (large) positive integer $n$, set $h_n(k)=nh(k/n)$ for any $k\in \{0,\dots ,n\}$. We consider a random walk $(S_k{)}_{k\ge 0}$ with i.i.d.\ centred increments having some finite exponential moments. We are interested in the event $\{S\ge h_n\}=\{S_k\ge h_n(k)\forall k\in \{0,\dots ,n\}\}$. It is well known that $P(S\ge h_n|S_0=0,S_n=\lceil h_n(n)\rceil )=e^{n{\int }_0^{1}I(h^{\prime }(s))ds+o(n)}$, 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 $e^{-\Theta (n^{1/3})}$. We then turn our attention to the conditional random walk measure $P_n^h=P(\cdot |S\ge h_n,S_0=0,S_n=\lceil h_n(n)\rceil )$. We prove that the one-point tails are of the form ${\mathbb{P}}_n^h(S_k\ge h_n(k)+t{n}^{1/3})=e^{-\Theta (t^{3/2})}$ for all $t<n^{\beta }$ for any $\beta \in (0,1/6)$. Moreover, we prove that, for any $r\ge 1$, $E_n^h((S_k-h_n(k){)}^r)=\Theta (n^{r/3})$ and ${\mathrm{Var}}_{P_n^h}(S_k)=\Theta (n^{2/3})$, for all $k$ far enough from $0$ and $n$. In addition, we show that ${\mathrm{Cov}}_{P_n^h}(S_k,S_{\ell })\le e^{-O(|\ell -k|/n^{2/3})}$ for all $k,\ell$ not too close to $0$ and $n$.