User:Lang.White/sandbox

Reciprocal Random Processes
Reciprocal random processes are a class of random processes defined on a one-dimensional index set, usually the closed interval [0,T] of the real line, or the subset {0,...,T} of the integers. All Markov processes are reciprocal but the converse is not true. Reciprocal processes are the one-dimensional analogue of Markov random fields in that they share the "nearest-neighbour" property. The process random variables can take values in the real line or in a finite set. In the latter case, the reciprocal process is sometime referred to as a reciprocal chain.

Reciprocal processes were first studied in the early part of the twentieth century by Bernstein and Shrödinger,. Schrödinger studied the behaviour of an electron on an interval of the real line, and described a method of construction of a reciprocal process to describe the probability density of the location of the electron. All these processes were actually Markov with prescribed marginal distributions on the end points. This study led to the necessary and sufficient conditions for a reciprocal process to be Markov. This involved the solution of two coupled functional equations, the solution of which was studied in more detail by Fortet. Some time later, Slepian applied the theory of reciprocal processes to study the problem of determining the probability density function of the first passage time of a particular (real index parameter) Gaussian process. . The construction of reciprocal processes and their relationship with Markov processes was studied in detail by Jamison. Stationary continuous parameter Gaussian reciprocal processes were also characterised by Jamison.

Towards the end of the twentieth century, there was renewed interest in reciprocal processes from theoretical and practical viewpoints. In a series of papers, B. Levy et al studied the continuous parameter, continuous range case and derived partial differential equations to describe the probability density function of the process. In contrast with the analogous equations for Markov process, (Fokker-Planck equations ) these equations are second-order.

In more recent times, there has been considerable attention paid to the case of discrete parameter reciprocal processes. Stochastic realisation and optimal smoothing for Gaussian discrete parameter processes was addressed in