Piecewise linear random paths on a plane and a central limit theorem

A piecewise linear random path from a sequence of line processes on a plane is constructed. It is shown that the entropy of such a path is maximized if the sequence of line processes are independent identically distributed, Poisson line processes. We also establish that the properly scaled (contracted) coordinates of a constantly moving object on a piecewise linear random path become asymptotically normal as time goes on if the sequence of line processes is independent identically distributed and satisfies certain conditions.