Spectral correlation characteristics, invariant under one-to-one delayless functional transformations of stochastic processes (Q1105911)

The cross concorrelation function \(K_{XY}(t_ 1,t_ 2)\) and the normalized concorrelation function \(\kappa_{XY}(t_ 1,t_ 2)\) of stochastic processes X(t) and Y(t) are defined as \[ K_{XY}(t_ 1,t_ 2)=E\{F_ X[X(t_ 1)]-E F_ X[X(t_ 1)]\}\{F_ Y[Y(t_ 2)]-E F_ Y[Y(t_ 2)]\}, \] \[ \kappa_{XY}(t_ 1,t_ 2)=K_{XY}(t_ 1,t_ 2)/\sqrt{K_{XX}(t_ 1,t_ 1)K_{YY}(t_ 2,t_ 2)}, \] where \(F_ X(x)=F_ X(x;t)\) is the distribution function of X(t). If X(t) and Y(t) are stationary and stationarily correlated processes one can introduce by means of the `concors' defined above, the conspectral power density. These characteristics do not change their values if X(t) and Y(t) undergo one-to-one delayless functional transformations. Some properties and practical examples of the utilization of concors (for the determination of the strength of the correlation between \(X(t_ 1)\) and \(Y(t_ 2)\), for the description of delay elements of nonlinear systems, for the simulation of random functions) are given.