Likelihood Ratios for Processes
As seen in the preceding work on inference theory of processes, likelihood ratios play a prominent role, particularly for the testing problems. Consequently the major part of this chapter will be devoted to finding densities for probability measures induced by broad classes of stochastic processes. These include processes with independent increments, jump Markov, and those that are infinitely divisible. Also considered for this work are diffusion types of processes and some applications. All these start with (and suggested by) Gaussian processes and so we establish results including dichotomy theorems as well as likelihood ratios for them under several different sets of conditions, together with a few stationary cases. As a motivation for (and also an interest in) the subject we start with a treatment of the important problem of (admissible) means of processes within their function space representations (and these means can be regarded as deterministic signals). The analysis presented in this chapter involves some interesting mathematical ideas, and the reader should persevere with patience, since a rich collection of problems, applications, and new directions are suggested in this work. We include many illustrations in the text, and some further results in the final complements section, often with hints.
KeywordsLikelihood Ratio Gaussian Process Gaussian Measure Reproduce Kernel Hilbert Space Independent Increment
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- Pitcher, T. S. Likelihood ratios of Gaussian processes,“ Ark. Mat. 4 (1959), 35–44.Google Scholar
- Kailath, T. The structure of Radon-Nikodÿm derivatives with respect to the Wiener and related measures,“ Ann. Math. Statist.,42 (1971), 1054–1067.Google Scholar
- Skorokhod, A. V. On admissible translations of measures in Hilbert space,“ Theor. Prob. Appl.,15 (1970), 557–580.Google Scholar
- Gikhman, I. I., and Skorokhod, A. V.“On the densities of probability measures in function spaces,” Russian Math Surveys21(6) (1966), 83–156. Google Scholar
- Choksi, J. R. “Inverse limits of measure spaces,” Proc. Lond. Math. Soc. (3) 8 (1958), 321–342.Google Scholar
- Brockett, P. L., and Tucker, H. G. Brillinger, D. R. “A conditional dichotomy theorem for stochastic processes with independent increments,” J. Multivar. Anal., 7 (1977), 13–27. Google Scholar
- Kakutani, S. On the equivalence of infinite product measures,“ Ann. Math.,49 (1948), 214–224.Google Scholar
- Krinik, A. Diffusion processes in Hilbert space and likelihood ratios,“ In Real and Stochastic Analysis,Wiley, New York (1986), 168–210. Kühn, T., and Liese, F.Google Scholar
- Basawa, I. V., and Prakasa Rao, B. L. S. Statistical Inference on Stochastic Processes, Academic Press, New York, 1980.Google Scholar