On a Functional Central Limit Theorem for Random Walks Conditioned to Stay Positive

Let ₖ: k 1\ be a sequence of i.i.d.rv with E(Xᵢ) = 0 and E(Xᵢ²) = σ², 0 < σ² < ∞. Set Sₙ = X₁ + ⋯ + Xₙ. Let Yₙ(t) be Sₖ/σ n¹/2 for $t = k/n$ and suitably interpolated elsewhere. This paper gives a generalization of a theorem of Iglehart which states weak convergence of Yₙ(t), conditioned to stay positive, to a suitable limiting process.