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On a limit behaviour of a random walk penalised in the lower half-plane
A. Pilipenkoab, O. O. Prykhodkoc a Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska str., 01601, Kyiv, Ukraine
b National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine
c National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Department of Physics and Mathematics, 03056, Kyiv, Ukraine, 37, Peremohy ave
Abstract:
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.
Citation:
A. Pilipenko, O. O. Prykhodko, “On a limit behaviour of a random walk penalised in the lower half-plane”, Theory Stoch. Process., 25(41):2 (2020), 81–88
Linking options:
https://www.mathnet.ru/eng/thsp320 https://www.mathnet.ru/eng/thsp/v25/i2/p81
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Abstract page: | 110 | Full-text PDF : | 24 | References: | 27 |
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