We consider the random elephant walk, where the increments depend on the whole history of the process. We calculate functions of expected value and variance of the process and see, that depending on the values of parameters, the process exhibits diffusive and superdiffusive behaviour. We discuss the convergence of the process and show, that the strong law of large numbers holds. In diffusive regime and also at the transition point, the process, when suitably normalized, converges to a normal random variable. However, this is not the case in superdiffusive regime, where we have convergence to a non-degenerate, yet not normal random variable.
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