Statistics of an autoregressive correlated random walk along a return path

A closed-form analytical solution is derived for the statistical outcome of a random walk along a return path. The random walk is generated from the cumulative sum of correlated samples in a Gaussian-distributed autoregressive sequence. The outcome exhibits a smaller variance compared with a one-way path of equivalent length due to cancellation of correlated steps along the return leg. Furthermore, the variance decreases towards zero as the correlation coefficient approaches unity. An example application for this general result is the modelling of cumulative errors in dead-reckoning navigation systems, e.g. Doppler velocity log-aided inertial navigation systems used commonly on underwater vehicles. In this particular application, it can be used to express and quantify the natural cancellation of correlated error components between subsequent opposing legs in a typical ‘lawnmower’ survey pattern.