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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{THSP_2020_25_2_a7,
     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/}
}
TY  - JOUR
AU  - A. Pilipenko
AU  - O. O. Prykhodko
TI  - On a limit behaviour of a random walk penalised in the lower half-plane
JO  - Teoriâ slučajnyh processov
PY  - 2020
SP  - 81
EP  - 88
VL  - 25
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a7/
LA  - en
ID  - THSP_2020_25_2_a7
ER  - 
%0 Journal Article
%A A. Pilipenko
%A O. O. Prykhodko
%T On a limit behaviour of a random walk penalised in the lower half-plane
%J Teoriâ slučajnyh processov
%D 2020
%P 81-88
%V 25
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a7/
%G en
%F THSP_2020_25_2_a7
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/

[1] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Cambridge University Press, Cambridge, 1987

[2] P. Billingsley, Convergence of Probability Measures, Wiley, 1999

[3] A. Gut, Probability: A Graduate Course, Springer, New York, 2013

[4] C. Hahn, Z. R. Kann, J. A. Faust, J. L. Skinner, G. M. Nathanson, “Super-Maxwellian helium evaporation from pure and salty water”, The Journal of Chemical Physics, 144(4):044707, January (2016)

[5] A. M. Johnson, D. K. Lancaster, J. A. Faust, C. Hahn, A. Reznickova, G. M. Nathanson, “Ballistic evaporation and solvation of helium atoms at the surfaces of protic and hydrocarbon liquids”, The Journal of Physical Chemistry Letters, 5:21 (October 2014), 3914–3918

[6] Z. R. Kann, J. L. Skinner, “Sub- and super-Maxwellian evaporation of simple gases from liquid water”, The Journal of Chemical Physics, 144:15:154701 (April 2016)

[7] J. L. Menaldi, “Stochastic variational inequality for reflected diffusion”, Indiana University mathematics journal, 32:5 (1983), 733–744

[8] A. Pilipenko, An introduction to stochastic differential equations with reflection, Potsdam:Universitatsverlag, 09, 2014

[9] A. Pilipenko, Yu. Prykhodko, “Limit behaviour of a simple random walk with non-integrable jump from a barrier”, Theory of Stochastic Processes, 19 (35):1 (2014), 52–61