Recognition of direction of point random shift accounting random rotation

The observation of the point on a plane is being made in order to determine the direction of its parallel shift. The point performs a complex motion which includes a parallel shift in a certain direction and rotation at a random angle. The observation begins at a random moment of time. Point coordinates are measured and recorded after each step of the random motion. Mathematical models have been constructed for the point motion and decisive rule of recognition. Expressions for calculating the probability of correct recognition of the parallel shift direction have been presented in the form of theorems.