On the second derivative of a Gaussian process envelope

In this paper we explore some dynamic characteristics of the envelope of a bandpass Gaussian process, which are of interest in wireless fading channels. Specifically, we show that unlike the first derivative, the second derivative of the envelope, which appears in a number of applications, does not exist in the traditional mean square sense. However, we prove that the envelope is twice differentiable almost everywhere (with probability one), if the power spectrum of the bandpass Gaussian process satisfies a certain condition. We also derive an integral-form for the probability density function of the second derivative of the envelope, assuming an arbitrary power spectrum.