On a limit behaviour of a random walk penalised in the lower half-plane

A. Pilipenko, O. O. Prykhodko
Theory of Stochastic Processes
Vol.25 (41), no.2, 2020, pp.81-88
We consider a random walk Ŝ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that {Ŝ(nt)/√n} has no weak limit in D([0,1]); alternatively, the weak limit is a reflected Brownian motion.
DOI: https://doi.org/10.37863/tsp-1140919749-78
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