Geodesic
On a limit behavior of a random walk with modifications upon each visit to zero
Teoriâ slučajnyh processov, Tome 22 (2017) no. 1, pp. 71-80.

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

We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are “small”, (ii) a linear motion with a random slope if modifications are “large”, and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are “moderate”.
Keywords: Invariance principle, self-interacting random walk, perturbed random walk. 
@article{THSP_2017_22_1_a6,
     author = {Andrey Pilipenko and Vladislav Khomenko},
     title = {On a limit behavior of a random walk with modifications upon each visit to zero},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {71--80},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2017_22_1_a6/}
}
TY  - JOUR
AU  - Andrey Pilipenko
AU  - Vladislav Khomenko
TI  - On a limit behavior of a random walk with modifications upon each visit to zero
JO  - Teoriâ slučajnyh processov
PY  - 2017
SP  - 71
EP  - 80
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2017_22_1_a6/
LA  - en
ID  - THSP_2017_22_1_a6
ER  - 
%0 Journal Article
%A Andrey Pilipenko
%A Vladislav Khomenko
%T On a limit behavior of a random walk with modifications upon each visit to zero
%J Teoriâ slučajnyh processov
%D 2017
%P 71-80
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2017_22_1_a6/
%G en
%F THSP_2017_22_1_a6
Andrey Pilipenko; Vladislav Khomenko. On a limit behavior of a random walk with modifications upon each visit to zero. Teoriâ slučajnyh processov, Tome 22 (2017) no. 1, pp. 71-80. http://geodesic.mathdoc.fr/item/THSP_2017_22_1_a6/

[1] A. N. Borodin, “An asymptotic behaviour of local times of a recurrent random walk with finite variance”, Theory Probab. Appl., 26:4 (1982), 758–772 | DOI | MR | Zbl

[2] D. Dolgopyat, “Central limit theorem for excited random walk in the recurrent regime”, ALEA Lat. Am. J. Probab. Math. Stat, 8 (2011), 259–268 | MR | Zbl

[3] W. Feller, An Introduction to Probability Theory and Its Applications, v. 1, J. Wiley $\$ sons, 1960 | MR

[4] I. I. Gikhman, A. V. Skorokhod, “On the densities of probability measures in function spaces”, Russian Mathematical Surveys, 21:6 (1966), 83–156 | DOI | MR

[5] J. M. Harrison, L. A. Shepp, “On skew Brownian motion”, Ann. Probab., 9 (1981), 309–313 | DOI | MR | Zbl

[6] A. M. Iksanov, A. Yu. Pilipenko, “A functional limit theorem for locally perturbed random walks”, Probab. Math. Statist., 36:2 (2016), 353-–368 | MR | Zbl

[7] E. Kosygina, M. P. Zerner, “Excited random walks: results, methods, open problems”, Bull. Inst. Math. Acad. Sin. (N.S.), 8:1, in a special issue in honor of S.R.S. Varadhan's 70th birthday (2013), 105–157 | MR | Zbl

[8] R. Liptser, A. N. Shiryaev, Statistics of random processes, Nauka, Moscow, 1974 (Russian) | MR

[9] R. A. Minlos, E. A. Zhizhina, “Limit diffusion process for a non-homogeneous random walk on a one-dimensional lattice”, Russ. Math. Surv., 52 (1997), 327–340 | DOI | MR | Zbl

[10] A. Pilipenko, Yu. Prykhodko, “On a limit behavior of a sequence of Markov processes perturbed in a neighborhood of a singular point”, Ukrainian Math. Journal, 67:4 (2015), 499–516 | DOI | MR | Zbl

[11] A. Yu. Pilipenko, Yu. E. Prykhodko, “Limit behavior of a simple random walk with non-integrable jump from a barrier”, Theor. Stoch. Proc., 19(35) (2014), 52–61 | MR | Zbl

[12] A. Pilipenko, L. Sakhanenko, “On a limit behavior of one-dimensional random walk with non-integrable impurity”, Theory of Stochastic Processes, 20(36):2 (2015), 97–104 | MR | Zbl

[13] O. Raimond, B. Schapira, “Excited Brownian motions as limits of excited random walks”, Probability Theory and Related Fields, 154:3-4 (2012), 875–909 | DOI | MR | Zbl

[14] A. N. Shiryaev, Probability, Graduate texts in mathematics, 95, 1996 | DOI | MR

[15] D. Szász, A. Telcs, “Random walk in an inhomogeneous medium with local impurities”, J. Stat. Physics, 26 (1981), 527–537 | DOI | MR | Zbl

[16] M. P. W. Zerner, “Recurrence and transience of excited random walks on $Z^d$ and strips”, Electron. Comm. Probab., 11:12 (2006), 118–128 | DOI | MR | Zbl