Markov and Conditionally Markov Processes: From Gaussian to Elliptical

Conditionally Markov (CM) processes, including the reciprocal processes as an important subclass, are gaining momentum as a generalization of the Markov process based on conditioning. This paper aims to extend the concept of the Gaussian CM process and some of its key results to the elliptical case. An elliptical process describes a stochastic process having jointly elliptically contoured distributions, which is the largest class of processes ensuring linearity of conditional expectation. However, it is shown in this paper that a nonsingular elliptical process is CM if and only if it is a Gaussian CM process. Towards the goal of extension, we first define a new property of processes, called weaker Markov, by relaxing the strict independence for the Markov property to a weaker condition of semi-independence. We then combine it with conditioning and the elliptical randomness to define the conditionally weaker Markov (CWM) elliptical process. The newly defined process is much larger than the Gaussian CM process, but can be characterized by a linear model of the same form as in the Gaussian case. Moreover, it is proven that the two processes also share almost the same results in optimal filtering and smoothing.