In this paper we consider a linear discrete time invariant system under random disturbances and with the nonzero initial condition. By the analogy with the anisotropic norm of a stochastic system the disturbance attenuation capabilities of system are quantified by the anisotropic generalized gain which is defined as the largest root mean square gain of the system with respect to a random input and the nonzero initial condition, anisotropy of which is bounded by a given nonnegative parameter a. A numerical example is given.