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

Kôhei Uchiyama ()
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Kôhei Uchiyama: Tokyo Institute of Technology

Journal of Theoretical Probability, 2020, vol. 33, issue 3, 1296-1326

Abstract: Abstract 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

Keywords: Random walk on the integer lattice; Conditioned to avoid a set; Third moment; Functional limit theorem; Killing on a finite set; Tightness of pinned walk; Tunneling; Primary 60G50; Secondary 60J45 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00908-x

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