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We analyse large deviations of time-averaged quantities in stochasticprocesses with long-range memory, where the dynamics at time t depends itselfon the value q_t of the time-averaged quantity. First we consider the elephantrandom walk and a Gaussian variant of this model, identifying two mechanismsfor unusual fluctuation behaviour, which differ from the Markovian case. Inparticular, the memory can lead to large deviation principles with reducedspeeds, and to non-analytic rate functions. We then explain how the mechanismsoperating in these two models are generic for memory-dependent dynamics andshow other examples including a non-Markovian symmetric exclusion process.

Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory