Here the procedure of Lyapunov functionals construction described in Chap. 1 is used for the investigation of asymptotic mean square stability of the illustrative scalar difference equation with constant coefficients. Using four different representations of the considered difference equation in the special auxiliary form different conditions for asymptotic mean square stability are obtained. One generalization are obtained also for p-stability with p=2k, k=1,2,… . The results of investigations are illustrated by figures with obtained stability regions.