Central Limit Theorem For The Excited Random Walk In Dimension $d\geq 2$

Abstract

We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension d ≥ 2.1991 Mathematics Subject Classification. 60K35, 60J10.

Monotonicity for excited random walk in high dimensions

“…Simulations [5] suggest that the limiting variance of the first coordinate is not monotone in the excitement parameter β in 2 dimensions. We expect that using the approach introduced in this paper, we can show that the variance is monotone decreasing in β when the dimension is taken sufficiently high.…”

“…Since the speed is known to exist [5], the following corollary is an easy consequence of [15, Propositions 3.1 and 6.1] together with Proposition 3.2, and the fact that a 6 < 1 since G * 2 5 < 5 2 /6 [9]. …”

“…[5]) and (3.20) where the right hand side which converges as n → ∞ since P(ω n ∈ ω n−1 ) is the µ n -measure of the event that the cookie at the origin is absent.…”

“…More recently, [5] explicitly proved a SLLN and established a functional central limit theorem in dimensions d ≥ 2. Included in [15] is an explicit representation of the drift in terms of the expansion coefficients.…”

Stochastic Sequences with a Regenerative Structure that May Depend Both on the Future and on the Past

“…[9], [11], and [16]); excited random walks (see, e.g. [4], [5], and [21]); modified random walks (see, e.g. [25]).…”