Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive

Let \ξₖ: k 1\ be a sequence of independent, identically distributed random variables with E\ξ₁\ = 0 and E\ξ₁²\ = σ², 0 < σ² < ∞. Form the random walk ₙ: n 0\ by setting S₀ = 0, Sₙ = ξ₁ + ⋯ + ξₙ, n 1. Let T denote the hitting time of the set (-∞, 0 by the random walk. The main result in this paper is a functional central limit theorem for the random functions Sₙₜ/σ n1/2, 0 t 1, conditional on $T > n$. 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\ξ₁\ = μ > 0, it is shown that the conditional (on $T > n$) and unconditional weak limit for (Sₙₜ - μ nt)/σ n1/2 is the same, namely, Brownian motion.