Local Central Limit Theorem for a Random Walk Perturbed in One Point
Article
First Online:
- 46 Downloads
Abstract
We consider a symmetric random walk on the ν-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction to diffusive behaviour appears in any dimension along with a long-range correction in the one-dimensional case.
Keywords
Local central limit theorems Inhomogeneous random walksMathematics Subject Classification (2010)
60F05 60G50 60J10Notes
Acknowledgements
The authors thank C. Boldrighini for suggesting the problem. R. L. is supported by the ERC grant 676675 FLIRT.
References
- 1.Boldrighini, C., Marchesiello, A., Saffirio, C.: Weak dependence for a class of local functionals of Markov chains on \(\mathbb {Z}^{d}\). Methods Funct. Anal. Topology 21, 302–314 (2015)MathSciNetzbMATHGoogle Scholar
- 2.Boldrighini, C., Pellegrinotti, A.: Random walk on \(\mathbb {Z}\) with one-point inhomogeneity. Markov Proc. Rel. Fields 18, 421–440 (2012)MathSciNetzbMATHGoogle Scholar
- 3.Szegö, G.: Orthogonal Polynomials, vol. XXIII. AMS Colloquium Publications, Providence (1933)Google Scholar
- 4.Korshunov, D.A.: Limit theorems for general markov chains. Siberian Math. J. 42(2), 301–316 (2001)MathSciNetCrossRefGoogle Scholar
- 5.Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
- 6.Minlos, R.A., Zhizhina, E.A.: Local limit theorem for a non homogeneous random walk on the lattice. Theory Probab. Appl. 39, 513–529 (1994)MathSciNetzbMATHGoogle Scholar
Copyright information
© Springer Nature B.V. 2019