The effect of the interaction between the amplitude and phase fluctuations of a laser wave on the laser emission linewidth is considered for the case when the single-mode laser-emission frequency does not coincide with the center of the homogeneously broadened gain profile. It is found that, when passing from a conventional laser to a sub-Poisson one with regular excitation, no narrowing of the line occurs. Therefore, simple considerations, according to which the reduction of fluctuations in the population difference should lead to decreasing fluctuations in the frequency pulling and, as a consequence, to decreasing linewidth of the laser emission, are not justified. The physical reasons why elimination of Poisson fluctuations in the pumping does not affect the phase-diffusion rate are discussed. When the laser action occurs outside the gain line center, the phase noise statistics remains Gaussian. This is supported by the form of the pseudocharacteristic function of the laser wave strength derived in the present work.