Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2
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Random Walks on Comb-Type Subsets of Z2 Endre Csáki1 · Antónia Földes2 Received: 28 March 2019 / Revised: 7 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We study the path behavior of the simple symmetric walk on some comb-type subsets of Z2 which are obtained from Z2 by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation results and discuss their consequences. Keywords Random walk · 2-dimensional comb · Strong approximation · 2-dimensional Wiener process · Oscillating Brownian motion · Laws of the iterated logarithm · Iterated Brownian motion Mathematics Subject Classification (2010) Primary 60F17 · 60G50 · 60J65; Secondary 60F15 · 60J10
1 Introduction An anisotropic walk is defined as a nearest-neighbor random walk on the square lattice Z2 of the plane with possibly unequal symmetric horizontal and vertical step probabilities, so that these probabilities depend only on the value of the vertical coordinate. More formally, consider the random walk {C(N ) = (C1 (N ), C2 (N )) ; N = 0, 1, 2, . . .} on Z2 with the transition probabilities
A. Földes: Research supported by a PSC CUNY Grant, No. 61520-0049.
B
Antónia Földes [email protected] Endre Csáki [email protected]
1
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest P.O.B. 127, 1364, Hungary
2
Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York 10314, USA
123
Journal of Theoretical Probability
P(C(N + 1) = (k + 1, j)|C(N ) = (k, j)) = P(C(N + 1) = (k − 1, j)|C(N ) = (k, j)) =
1 − pj, 2
P(C(N + 1) = (k, j + 1)|C(N ) = (k, j)) = P(C(N + 1) = (k, j − 1)|C(N ) = (k, j)) = p j ,
(1.1)
for (k, j) ∈ Z2 , N = 0, 1, 2, . . . with 0 < p j ≤ 1/2 and min j∈Z p j < 1/2. Unless otherwise stated, we assume also that C(0) = (0, 0). In the present paper, we are interested in a special type of this anisotropic walk. We only want to consider walks for which p j in (1.1) is either 1/2 or 1/4. In particular, for such walks we consider an arbitrary subset B of the integers on the y-axis and remove from the two-dimensional integer lattice all the horizontal lines which do not belong to the y-levels in B. Denote this lattice by C2 = C2 (B). The transition probabilities throughout this paper are p y = P(C(N + 1) = (x ± 1, y) | C(N ) = (x, y)) 1 , if y ∈ B 4 1 p y = P(C(N + 1) = (x, y ± 1) | C(N ) = (x, y)) = , if y ∈ / B, 2 = P(C(N + 1) = (x, y ± 1) | C(N ) = (x, y)) =
(1.2)
A compact way of describing the just introduced transition probabilities for this simple random walk C(N ) on C2 (B) is via defining p(u, v) := P(C(N + 1) = v | C(N ) = u) =
1 , deg(u)
(1.3)
for locations u and v that are neighbors on C2 (B), where deg(u) is the number of neighbors of u, otherwise p(u, v) := 0. Clearly when B = {0}, we get the two-dimensional comb which inspired our choice for the name of these particular anisotropic walks. We are interested in the case when Bn := B ∩ [−n, n], a
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