This paper considers scale invariance of statistical image models. We study statistical scale invariance of the covariance structure of jet space under scale space blurring and derive the necessary structure and conditions of the jet covariance matrix in order for it to be scale invariant. As part of the derivation, we introduce a blurring operator A_t that acts on jet space contrary to doing spatial filtering and a scaling operator S_s. The stochastic Brownian image model is an example of a class of functions which are scale invariant with respect to the operators A_t and S_s. This paper also includes empirical results where we estimate the scale invariant jet covariance of natural images and show that it resembles that of Brownian images.