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

TL;DR

This paper analyzes the behavior of a conditioned random walk constrained to stay above a curved boundary with negative curvature, revealing precise asymptotics and fluctuation properties related to curvature effects.

Contribution

It provides new asymptotic estimates for the probability and fluctuations of a random walk conditioned to stay above a curved boundary with negative curvature.

Findings

Leading correction to probability is of order e^{- ext{const} imes n^{1/3}}

Conditional tail probabilities decay as e^{- ext{const} imes t^{3/2}}

Variance of the walk's position scales as n^{2/3}

Abstract

Let $h:[0,1]\to\mathbb{R}$ be $C^2$ and such that $\sup_{[0,1]} h''<0$. For a (large) positive integer $n$, set $h_n(k) = n h(k/n)$ for any $k\in\{0,\dots,n\}$. 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 $\{S\geq h_n\} = \{S_k\geq h_n(k)\;\forall k\in\{0,\dots,n\}\}$. It is well known that $P(S\geq h_n \,|\, S_0=0,\, S_n=\lceil h_n(n) \rceil) = e^{n\int_0^1 I(h'(s)) \,ds + o(n)}$, 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 $e^{-\Theta(n^{1/3})}$. We then turn our attention to the conditional random walk measure $P^h_n = P(\cdot \,|\, S\geq h_n, S_0=0, S_n=\lceil h_n(n) \rceil)$. We prove that the one-point tails are of the form $\mathbb{P}_n^h (S_k \geq h_n(k) + t n^{1/3} ) = e^{-\Theta(t^{3/2})}$ for all $t<n^\beta$ for any $\beta\in (0,1/6)$. Moreover, we prove that, for any $r\geq 1$, $E_n^h((S_k-h_n(k))^r) = \Theta(n^{r/3})$ 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 $\mathrm{Cov}_{P_n^h}(S_k,S_\ell) \leq e^{-O(|\ell-k|/n^{2/3})}$ for all $k,\ell$ not too close to $0$ and $n$.