"Robotic" estimation: The inefficiency of random-walk sampling

Pattern recognition often involves estimation of statistics of a data ensemble by taking samples of the data and using sample statistics as estimates of the ensemble statistics. Ideally, the samples should be chosen randomly from the ensemble. In some situations, however, random sampling may not be practical. For example, if a robot is required to obtain samples of its environment, it would be inefficient for the robot to go to a sequence of randomly chosen locations to collect samples. In moving through the environment, the robot must follow a continous path, and it can obtain large numbers of samples as it moves along the path. This "robotic sampling" process can be made (somewhat) random by letting the path be a random walk through the environment. Unfortunately, if successive samples along the path are correlated, taking samples along the path is less efficient (from a sampling-theoretic standpoint) than taking random samples. This paper studies the inefficiency of "robotic" estimation, based on a random-walk path, relative to estimation based on random sampling.