Geodesic
On a limit behaviour of a random walk penalised in the lower half-plane
Teoriâ slučajnyh processov, Tome 25 (2020) no. 2, pp. 81-88

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We consider a random walk $\tilde S$ 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 $\{\frac{1}{\sqrt n} \tilde S(nt)\}$ has no weak limit in $\mathcal D$; alternatively, the weak limit is a reflected Brownian motion.
Keywords: Invariance principle, Reflected Brownian motion. 
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     author = {A. Pilipenko and O. O. Prykhodko},
     title = {On a limit behaviour of a random walk penalised in the lower half-plane},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {81--88},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a7/}
}
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A. Pilipenko; O. O. Prykhodko. On a limit behaviour of a random walk penalised in the lower half-plane. Teoriâ slučajnyh processov, Tome 25 (2020) no. 2, pp. 81-88. http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a7/