Coordinate-wise Elephant Random Walk

We introduce a coordinate-wise version of the elephant random walk on the $k$-dimensional discrete hypercube $Q_k=\{-1,1\}^k$.

At each global time step, one coordinate is selected uniformly at random and updated according to an elephant-type memory rule using only the past values of that coordinate.

The resulting process is a nearest-neighbor walk with possible holding on the hypercube, but it is not Markovian on $Q_k$ because the transition probabilities depend on coordinate-wise empirical histories.

We show that, when all memory parameters satisfy $p_i<1$, the coordinate-wise memory biases vanish almost surely.

Consequently, the time-dependent transition kernels of the walk are asymptotically close in total variation to the memoryless coordinate-refresh kernel.

Using this perturbation argument and the uniform ergodicity of the refresh chain, we prove a weak law of large numbers for bounded observables.

We then study the Doob martingale associated with a bounded observable and prove a martingale central limit theorem and functional central limit theorem.

The limiting variance is determined by the refresh kernel and the uniform measure on the hypercube, and is completely independent of the memory parameters $p_1,p_2,\dots,p_k.$