Gaussian heat kernel asymptotics for conditioned random walks

Consider a random walk $S_n={\sum }_{i=1}^n{X}_i$ with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order $2+\delta$ for some $\delta >0$. For any starting point $x\in \mathbb{R}$, let ${\tau }_x=\inf \{k\ge 1:x+S_k<0\}$ denote the first time when the random walk $x+S_n$ exits the half-line $[0,\mathrm{\infty })$. We investigate the uniform asymptotic behavior over $x\in \mathbb{R}$ of the persistence probability $\mathbb{P}({\tau }_x>n)$ and the joint distribution $\mathbb{P}(x+S_n\le u,{\tau }_x>n)$, for $u\ge 0$, as $n\to \mathrm{\infty }$. New limit theorems for these probabilities are established based on the heat kernel approximations. Additionally, we evaluate the rate of convergence by proving Berry-Esseen type bounds.