We consider the class of simple random walks or birth and death chains on the nonnegative integers. The set of return probabilities Pn00, n [greater-or-equal, slanted] 0, uniquely determines the spectral measure of the process. We characterize the class of simple random walks with the same...

In a recent paper, Mokkadem (1997. Stoch. Proc. Appl. 72, 145-149) derived a simple test for randomness against ARMA alternatives. In this note we consider a transformation of the corresponding statistic and present an alternative proof of this result. Through this approach it is demonstrated...