Fisher information of correlated stochastic processes

Many real-world tasks include some kind of parameter estimation, i.e., determination a encoded in probability distribution. Often, such distributions arise from stochastic processes. For stationary process with temporal correlations, the random variables that constitute it are identically distributed but not independent. This is case, for instance, quantum continuous measurements. In this paper we prove two fundamental results concerning estimation parameters memoryful process. First, show processes finite Markov order, Fisher information always asymptotically linear number outcomes, and determined by conditional distribution process' order. Second, suitable examples correlations do necessarily enhance metrological precision. fact, unlike entropic quantities, general nothing can be said about sub- or super-additivity joint information, presence correlations. We discuss how type affects scaling. then apply these to case thermometry on spin chain.