We address anomalous transport phenomena in arrays of semiconductor nanocrystals (quantum dots): Transient power-law decay of current as a response to a step in large bias voltage applied across the array, as well as memory effects observed after successive applications of the bias voltage. A novel phenomenological model of transport in such systems is proposed, capable of rationalizing both anomalous transport and memory. The model describes electron transport by a stationary Levy process of transmission events and therefore requires no time dependence of system properties. The long tail in the waiting time distribution gives rise to a nonstationary response in the presence of a voltage pulse. Noise measurements agree well with the predicted non-Poissonian fluctuations in current. We briefly discuss possible microscopic mechanisms that could cause the anomalous statistics in transmission.