## Electronic Journal of Probability

### Random walks in a moderately sparse random environment

#### Abstract

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_{n})_{n\in \mathbb{N} \cup \{0\}}$ in a sparse random environment $(S_{k},\lambda _{k})_{k\in \mathbb{Z} }$ is a nearest neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z} \setminus \{\ldots ,S_{-1},S_{0}=0,S_{1},\ldots \}$ and jumps to the right (left) with the random probability $\lambda _{k+1}$ ($1-\lambda _{k+1}$) from the point $S_{k}$, $k\in \mathbb{Z}$. Assuming that $(S_{k}-S_{k-1},\lambda _{k})_{k\in \mathbb{Z} }$ are independent copies of a random vector $(\xi ,\lambda )\in \mathbb{N} \times (0,1)$ and the mean $\mathbb{E} \xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_{n}$, properly normalized and centered, as $n\to \infty$. While the case $\xi \leq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} \xi =\infty$ (strong sparsity) will be analyzed in a forthcoming paper.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 69, 44 pp.

Dates
Accepted: 7 June 2019
First available in Project Euclid: 28 June 2019

https://projecteuclid.org/euclid.ejp/1561687602

Digital Object Identifier
doi:10.1214/19-EJP330

Mathematical Reviews number (MathSciNet)
MR3978219

Zentralblatt MATH identifier
07089007

#### Citation

Buraczewski, Dariusz; Dyszewski, Piotr; Iksanov, Alexander; Marynych, Alexander; Roitershtein, Alexander. Random walks in a moderately sparse random environment. Electron. J. Probab. 24 (2019), paper no. 69, 44 pp. doi:10.1214/19-EJP330. https://projecteuclid.org/euclid.ejp/1561687602

#### References

• [1] Anderson, K. K. and Athreya, K. B.: A note on conjugate $\Pi$-variation and a weak limit theorem for the number of renewals. Statist. Probab. Lett. 6, (1988), 151–154.
• [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L.: Regular variation. Cambridge University Press, 1989.
• [3] Bouchet, É., Sabot, C. and dos Santos, R. S.: A quenched functional central limit theorem for random walks in random environments under $(T)_{\gamma }$. Stoch. Proc. Appl. 126(4), (2016), 1206–1225.
• [4] Buraczewski, D., Damek, E. and Mikosch, T.: Stochastic models with power-law tails. The equation $X=AX+B$. Springer Series in Operations Research and Financial Engineering. Springer, 2016.
• [5] Buraczewski, D. and Dyszewski, P.: Precise large deviations for random walk in random environment. Electron. J. Probab. 23(114), (2018), 1–26.
• [6] Buraczewski, D., Dyszewski, P., Iksanov, A. and Marynych, A.: Random walks in a strongly sparse random environment, 2019. arXiv:1903.02972
• [7] Comets, F., Gantert, N. and Zeitouni, O.: Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118(1), (2000), 65–114.
• [8] Damek, E. and Kolodziejek, B.: A renewal theorem and supremum of a perturbed random walk. Electron. Commun. Probab. 23(82), (2018), 1–13.
• [9] Dembo, A., Peres, Y. and Zeitouni, O.: Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181(3), (1996), 667–683.
• [10] Denisov, D., Foss, S. and Korshunov, D.: Asymptotics of randomly stopped sums in the presence of heavy tails. Bernoulli 16(4), (2010), 971–994.
• [11] Dolgopyat, D. and Goldsheid, I.: Quenched limit theorems for nearest neighbour random walks in 1D random environment. Comm. Math. Phys. 315(1), (2012), 241–277.
• [12] Enriquez, N. I., Sabot, C. and Zindy, O.: Limit laws for transient random walks in random environment on $\mathbb{Z}$. Annales de l’institut Fourier 59, (2009), 2469–2508.
• [13] Feller, W.: Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67(1), (1949), 98–119.
• [14] Feller. W.: An introduction to probability theory and its applications. 2nd edition. Wiley, 1971.
• [15] Gantert, N. and Zeitouni, O.: Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194(1), (1998), 177–190.
• [16] Goldie, C. M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), (1991), 126–166.
• [17] Grincevičius, A. K.: The continuity of the distribution of a certain sum of dependent variables that is connected with independent walks on lines. Teor. Verojatnost. i Primenen. 19, (1974), 163–168.
• [18] Grincevičius, A. K.: On a limit distribution for a random walk on lines. Litovsk. Mat. Sb. 15(4), (1975), 79–91.
• [19] Greven, A. and den Hollander, F.: Large deviations for a random walk in random environment. Ann. Probab. 22(3), (1994), 1381–1428.
• [20] Grey, D. R.: Regular variation in the tail behaviour of solutions of random difference equations. Ann. Appl. Probab. 4(1), (1994), 169–183.
• [21] Gut, A.: Stopped random walks: limit theorems and applications. 2nd edition. Springer, 2009.
• [22] Harris, T. E.: First passage and recurrence distributions. Trans. Amer. Math. Soc. 73(3), (1952), 471–486.
• [23] Iksanov, A.: Renewal theory for perturbed random walks and similar processes. Birkhäuser, 2016.
• [24] Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, (1973), 207–248.
• [25] Kesten, H.: The limit distribution of Sinaĭ’s random walk in random environment. Phys. A 138(1-2), (1986), 299–309.
• [26] Kesten, H., Kozlov, M. V. and Spitzer, F.: A limit law for random walk in a random environment. Compositio Math. 30, (1975), 145–168.
• [27] Key, E. S.: Limiting distributions and regeneration times for multitype branching processes with immigration in a random environment. Ann. Probab. 15(1), (1987), 344–353.
• [28] Korshunov, D. A.: An analog of Wald’s identity for random walks with infinite mean. Siberian Math. J. 50(4), (2009), 663–666.
• [29] Kozlov, M. V.: Random walk in a one-dimensional random medium. Theory Probab. Appl. 18(2), (1974), 387–388.
• [30] Matzavinos, A., Roitershtein, A. and Seol, Y.: Random walks in a sparse random environment. Electron. J. Probab. 21, (2016), paper no. 72.
• [31] Meyer, P.-A.: Probability and potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966.
• [32] Pakes, A. G.: Further results on the critical Galton–Watson process with immigration. J. Austral. Math. Soc. 13, (1972), 277–290.
• [33] Pisztora, A. and Povel, T.: Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab. 27(3), (1999), 1389–1413.
• [34] Pisztora, A., Povel, T. and Zeitouni, O.: Precise large deviation estimates for a one-dimensional random walk in a random environment. Probab. Theory Related Fields 113(2), (1999), 191–219.
• [35] Sinaĭ, Ya. G.: The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatnost. i Primenen. 27(2), (1982), 247–258.
• [36] Solomon, F.: Random walks in a random environment. Ann. Probab. 3, (1975), 1–31.
• [37] Sznitman, A. and Zerner, M.: A law of large numbers for random walks in random environment. Ann. Probab. 27(4), (1999), 1851–1869.
• [38] Varadhan, S. R. S.: Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56(8), (2003), 1222–1245.
• [39] Zerner, M. P. W.: Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26(4), (1998), 1446–1476.
• [40] Zeitouni, O.: Random Walks in Random Environment. XXXI Summer School in Probability, (St. Flour, 2001). Lecture Notes in Math., 1837, Springer, 193–312, 2004.