Multidimensional Walks with Random Tendency
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Multidimensional Walks with Random Tendency Manuel González-Navarrete1 Received: 27 February 2020 / Accepted: 31 July 2020 / Published online: 26 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we obtain a functional limit theorem to Gaussian vectors. In superdiffusive regime, we obtain strong convergence to a non-Gaussian random vector and characterize its moments. Keywords Random walk · Diffusive regime · Superdiffusive regime · Opinion model
1 Introduction The one-dimensional elephant random walk (ERW) was introduced in [29] (see also [22]). It can be represented by a sequence {X 1 , X 2 , . . .} where n X i ∈ {−1, +1}, for all i ≥ 1. The position of the elephant at time n is given by Sn = i=1 X i , and S0 = 0. For the ERW model, it is supposed that the elephant remembers its full history and makes its (n + 1)-step by choosing t ∈ {1, . . . , n} uniformly at random and then X t , with probability p, X n+1 = (1) −X t , with probability 1 − p. where p ∈ [0, 1] is a parameter. In this sense, let denote N (n, +1) = #{i ∈ 1, 2, . . . , n : X i = +1}, the number of +1 steps until the n-step, and N (n, −1) analogously defined. The position of the elephant can be rewritten as Sn = N (n, +1) − N (n, −1). Therefore, the conditional probability of (n + 1)-step in direction +1 is given by N (n, +1) N (n, −1) + (1 − p) , (2) n n where n = N (n, +1) + N (n, −1). The probability (2) was used in [4] to introduce a relation with the classical Pólya urn process [26]. The evolution of a Pólya urn is stated as follows. An P(X n+1 = 1|N (n, +1), N (n, −1)) = p
Communicated by Gregory Schehr.
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Manuel González-Navarrete [email protected] Departamento de Estadística, Universidad del Bío-Bío, Avda. Collao 1202, CP 4051381 Concepción, Chile
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urn starts with an initial quantity of R0 red and B0 blue balls and draws are made sequentially. After each draw, the ball is replaced and another ball of the same color is added to the urn. Let us consider the following notation: the urn is represented by the two-dimensional vector (Rn , Bn ), where Rn and Bn represent the number of red and blue balls at time n ∈ N, respectively. The relation established in [4] considers N (n, +1) = Rn , N (n, −1) = Bn and the evolution in the Pólya urn is modified in the sense that, at each draw the same color is used with probability p and the opposite color with 1 − p. This fact allowed [4] to prove the existence of a transition from diffusive to super-diffusive behaviours for Sn , with critical pc = 43 (also proved by [12]). That is, the mean squared displacement is a linear function of time in the diffusive case ( p < pc ), but is given by a power law in the super-diffusive regime ( p > pc ). Analogous results have been
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