The enhanced strong invariance principle for the elephant random walk

The elephant random walk (ERW) is a discrete-time random walk on Z with a memory about the whole past. It has been shown that the asymptotic behavior of the ERW depends heavily on a memory parameter 0 <= p <= 1. In this work, we establish the strong invariance principle for the ERW in the diffusive regime 0 <= p < 3/4 and the critical regime p = 3/4, which enhances the known results of Coletti, Gava and Schutz.