Conditioned Stochastic Processes for Conditional Random Fields

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Abstract
Analytical development is presented for the theory of conditional random fields involving conditioning deterministic time functions. After discussion of their basic concept and their engineering significance, the probability distribution of the Fourier coefficients for conditioned stochastic processes is derived. Its physical interpretation is presented in terms of harmonic amplitudes and phase angles. On this basis, solutions are obtained for the time‐varying mean values and the variances of conditioned stochastic processes as well as their first‐passage probabilities. Numerical simulation of the conditional random fields is also performed for assumed power spectral density and coherence functions. These results are discussed in terms of the probability theory and engineering application. Specifically, effects of coherency and the number of conditioning deterministic time functions are examined.