How two elephants can learn from each other

Abstract

We consider a two-elephant walking model in which the elephants interact dynamically. At each time step, each elephant determines its next move randomly based on its partner's past movements. We show that the asymptotic behavior of the elephants mainly depends on the sign and the absolute value of the product of their reinforcement parameters. In various regimes, we establish the law of large numbers and the central limit theorem. Our proofs are based on a connection to the random recursive trees and employ stochastic approximation techniques and martingale methods.