Local limit theorems for conditioned random walks by the heat kernel approximation

We study the random walk $(S_n)_{n\geq 1}$ with independent and identically distributed real-valued increments having zero mean and an absolute moment of order $2 + \delta$ for some $\delta > 0$. For any starting point $x \in \mathbb{R}$, let $\tau_x = \inf\{k \geq 1 : x + S_k < 0\}$ denote the first exit time of the random walk $x + S_n$ from the half-line $[0, \infty)$. In the previous work [25], we established a Gaussian heat kernel approximation for both the persistence probability $\mathbb{P}(\tau_x > n)$ and the joint distribution $\mathbb{P}(x + S_n \leq \cdot, \tau_x > n)$, uniformly over $x \in \mathbb{R}$ as $n \to \infty$. In this paper, we leverage these results to establish a novel conditioned local limit theorem for the walk $(x + S_n)_{n \geq 1}$. For $\mathbb{Z}$-valued random walks, we prove that the joint probability $\mathbb{P}(x + S_n = y, \tau_x > n)$ is uniformly approximated by a distribution governed by the Gaussian heat kernel over all $x, y \in \mathbb{Z}$ as $n \to \infty$. Our new asymptotic unifies into a single comprehensive formula the classical local limit theorem by Caravenna [6], as well as various results relying on specific assumptions on $x$ and $y$. As a corollary, we obtain a new uniform-in-$x$ asymptotic formula for the local probability $\mathbb{P}(\tau_x = n)$. We also extend our analysis to non-lattice random walks.