  Elephant random walk is a special type of random walk that incorporates the
memory of the past to determine its future steps. The probability of this walk
taking a particular step (+1 or -1) at a time point, conditioned on the entire
history, depends on a linear function of the proportion of that step till that
time point. In this work, we investigate how the dynamics of the random walk
will change if we replace this linear function by a generic map satisfying some
analytic conditions. We also propose a new model, called the multidimensional
generalized elephant random walk, that incorporates several variants of
elephant random walk in one and higher dimensions and their generalizations
thereof. Using tools from the theory of stochastic approximation, we derive the
asymptotic behavior of our model leading to newer results on the phase
transition boundary between diffusive and superdiffusive regimes. We also
mention a few open problems in this context.
