The Gamma Variate with Random Shape Parameter and Some Applications

This letter provides a new outlook at the Gamma distribution and its involvement in the performance analysis of digital communications over wireless fading channels. Whereas the Gamma distribution is usually regarded as a two-parameter distribution with a scale and shape factors that are deterministic parameters, we investigate the implications and insight that can be drawn from considering a Gamma variate with a random shape factor. Applications of such a distribution in the context of information transmission over a wireless fading channel are provided and novel average symbol error probability (SEP) expressions for single and multi-channel reception are derived involving the Laplace transform of the new distribution. We analytically prove and verify via numerical simulations that the average SEP results induced by the random shape parameter are lower bounded by those obtained using a deterministic fading severity whose value equals the expected value of the random shape parameter.