Intensity fluctuations arising from a random walk in 2α dimensions are governed by the gamma-distribution of parameter α. When 2α is fractional, conventional methods of calculating statistical properties of the integrated intensity, based on a gaussian noise theory approach, can no longer be used. In this paper the single interval statistics of an integrated gamma-distributed variable with lorentzian spectrum are derived directly from a rate equation for the process. The results include, as a special case, the integrated photon statistics predicted by a model for single-mode thermal radiation in a cavity.