Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive

TLDR: In this article, it was shown that the conditional and unconditional weak limit for a sequence of independent, identically distributed random variables is the same as that of the standard Brownian motion.

Abstract: Let $\{\xi_k: k \geqq 1\}$ be a sequence of independent, identically distributed random variables with $E\{\xi_1\} = 0$ and $E\{\xi_1^2\} = \sigma^2, 0 n$. The limit process, $W^+$, is identified in terms of standard Brownian motion. Similar results are obtained for random partial sums and renewal processes. Finally, in the case where $E\{\xi_1\} = \mu > 0$, it is shown that the conditional (on $T > n$) and unconditional weak limit for $(S_{\lbrack nt\rbrack} - \mu nt)/\sigma n^{\frac{1}{2}}$ is the same, namely, Brownian motion.