Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set

This paper concerns a scaling limit of a one-dimensional random walk $$S^x_n$$ S n x started from x on the integer lattice conditioned to avoid a non-empty finite set A, the random walk being assumed to be irreducible and have zero mean. Suppose the variance $$\sigma ^2$$ σ 2 of the increment law is finite. Given positive constants b, c and T, we consider the scaled process $$S^{b_N}_{[tN]}/\sigma \sqrt{N}$$ S [ t N ] b N / σ N , $$0\le t \le T$$ 0 ≤ t ≤ T , started from a point $$b_N \approx b\sqrt{N}$$ b N ≈ b N conditioned to arrive at another point $$\approx -\,c\sqrt{N}$$ ≈ - c N at $$t=T$$ t = T and avoid A in between and discuss the functional limit of it as $$N\rightarrow \infty $$ N → ∞ . We show that it converges in law to a continuous process if $$E[|S_1|^3; S_1